On hereditarily reversible spaces

Topology and its Applications 225 (2017) 53–66

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Topology and its Applications www.elsevier.com/locate/topol

Virtual Special Issue – Proceedings on the International Conference on Set-Theoretic Topology and its Applications, Yokohama 2015

On hereditarily reversible spaces Vitalij A. Chatyrko a,∗ , Sang-Eon Han b , Yasunao Hattori c a

Department of Mathematics, Linkoping University, 581 83 Linkoping, Sweden Department of Mathematics Education, Institute of Pure and Applied Mathematics, Chonbuk National University, Jeonju-City Jeonbuk, 561-756, Republic of Korea c Department of Mathematics, Shimane University, Matsue, Shimane, 690-8504, Japan b

a r t i c l e

i n f o

Article history: Received 21 December 2015 Received in revised form 29 July 2016 Accepted 10 August 2016 Available online 25 April 2017 MSC: 54A10

a b s t r a c t In [15] Rajagopalan and Wilansky called a space reversible if each continuous bijection of the space onto itself is a homeomorphism. They called also a space hereditarily reversible if each its subspace is reversible. We characterize the hereditarily reversible spaces in several classes of topologicals spaces, in particular, in the class of Hausdorff spaces of the first countability and in some subclass of the class of locally finite T0 -spaces relevant to digital topology. Besides we suggest different examples of (non-)reversible and (non-)hereditarily reversible spaces. © 2017 Elsevier B.V. All rights reserved.

Keywords: Continuous bijection Reversible space Hereditarily reversible space Stone–Cech compactification of natural numbers Locally finite T0 -space

1. Introduction In [15] Rajagopalan and Wilansky called a space X reversible if each continuous bijection of the space onto itself is a homeomorphism, otherwise X is called non-reversible. It is easy to see that the property is topological, i.e. if two topological spaces are homeomorphic and one of them is reversible then the other one is also reversible. Simple examples of reversible spaces are discrete spaces Dκ of cardinality κ ≥ 0 (set D0 = ∅), compact Hausdorff spaces, Euclidean spaces Rn , finite spaces. Let us note that a space (X, τ ) is non-reversible iff there exists a strictly strongly (resp. weakly) topology τs (resp. τw ) on the set X such that (X, τ ) is homeomorphic to (X, τs ) (resp. (X, τw )). * Corresponding author. E-mail addresses: [email protected] (V.A. Chatyrko), [email protected] (S.-E. Han), [email protected] (Y. Hattori). http://dx.doi.org/10.1016/j.topol.2017.04.010 0166-8641/© 2017 Elsevier B.V. All rights reserved.

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The topological union Nℵ0 = Dℵ0 ⊕ cDℵ0 of Dℵ0 and cDℵ0 , where cDℵ0 is a one-point compactification of Dℵ0 , is non-reversible as well as the subspace Q (resp. P) of rational (resp. irrational) numbers of the real line R. One observes easily an irregular behavior of the reversibility under simple operations (as topological union and product). Thus the construction/identification of (non-)reversible spaces is one of the main subjects of this theory. Besides [15] many examples of both types one can find in [7,3,4,13] etc. In [15] Rajagopalan and Wilansky considered also hereditarily reversible spaces. (A space is hereditarily reversible if each its subspace is reversible.) They notated that an identification of hereditarily reversible spaces appeared difficult. Besides trivial examples of hereditarily reversible spaces as (a) any discrete space, (b) any space with only finitely many open sets, in particular, any finite space, (c) any space with cofinite topology, Rajagopalan and Wilansky suggested only ‘one’ example of a hereditarily reversible non-discrete Hausdorff space, namely: a subspace Vx = {x} ∪ N of the Stone–Cech compactification βN, where N is the space of natural numbers and x ∈ βN \ N. In this paper we will look for hereditarily reversible spaces (both Hausdorff and non-Hausdorff) and their characterizations. 2. Hereditarily reversible Hausdorff spaces of the countable character Let X be a space and p ∈ X. We denote by χ(p, X) the character of X at the point p and by χ(X) the character of X. Theorem 2.1. Let X be a infinite Hausdorff space and χ(X) ≤ ℵ0 . If X is neither homeomorphic to Dκ for any κ ≥ ℵ0 nor cDℵ0 then X contains a copy of Nℵ0 and hence X is not hereditarily reversible. Proof. Since the space X is not discrete, there is a limit point p of X, i.e. p ∈ ClX (X \ {p}). Case 1. For each open nbd W of p we have |X \ W | < ℵ0 . This implies that the subspace X \ {p} of X is homeomorphic to Dℵ0 and the space X is homeomorphic to cDℵ0 . We have a contradiction. Case 2. There exists an open nbd W of p such that |X \ W | ≥ ℵ0 . Note that χ(p, X) = ℵ0 . Let {Bn : n = 1, 2, . . . } be a base at p. We can assume that Bn+1 ⊆ Bn for each n ≥ 1. There exists Bn1 such that Bn1 ⊆ W . Choose a point x1 ∈ Bn1 \ {p}. Since X is Hausdorff, there exist open disjoint subsets O1 and V1 of Bn1 such that p ∈ O1 and x1 ∈ V1 . Choose n2 > n1 such that Bn2 ⊆ O1 , and then x2 ∈ Bn2 \ {p}, disjoint open subsets O2 and V2 of Bn2 such that p ∈ O2 and x2 ∈ V2 . Continuing as above we get a sequence n1 < n2 < . . . of integers, a sequence of points {xn }∞ n=1 , ∞ W two sequences {Oi }∞ and {V } of open sets. It is easy to see that the subspace Y = {p} ∪ {x : n ≥ 1} i i=1 n p i=1 of X is homeomorphic to cDℵ0 . Case 2.a. Each point of X \ W is not a limit point in X. So each point of X \ W is an open subset of X. Hence the set X \ W is clopen in X and the subspace X \ W is homeomorphic to Dτ for τ = |X \ W | ≥ ℵ0 . Choose any infinite countable subset Z of X \ W . Note that the subspace YpW ∪ Z of X is homeomorphic to Nℵ0 . Case 2.b. The set X \ W contains a limit point q of X.

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Since X is Hausdorff, there are disjoint open subsets Wp and Wq of X such that p ∈ Wp and q ∈ Wq . W W W W Similarly as above choose subsets Yp p of Wp and Yq q of Wq . Note that the subspace Yp p ∪ Yq q of X is homeomorphic to cDℵ0 ⊕ cDℵ0 . Hence X contains a subset which is homeomorphic to Nℵ0 . 2 Since each discrete space is hereditarily reversible as well as the space cDℵ0 , we have the following result. Corollary 2.1. (i) An infinite Hausdorff space X with χ(X) ≤ ℵ0 is not hereditarily reversible iff X contains a copy of Nℵ0 . (ii) The only hereditarily reversible Hausdorff spaces X with χ(X) ≤ ℵ0 are spaces either homeomorphic to Dκ for some κ ≥ 1 or cDℵ0 . Let us show a difference between the reversible spaces and the hereditarily reversible ones by cardinality. Recall (cf. [17]) that if two topological spaces are homeomorphic one says that they belong to the same topological type. Denote by Ms (resp. Mc ) the class of infinite separable (resp. countable) metrizable spaces. Thus Corollary 2.1 (ii), in particular, states that there are only two topological types of hereditarily reversible spaces in Ms (resp. Mc ), namely: Dℵ0 and cDℵ0 . For the reversible spaces in Ms and Mc the situation is different. Let K be a class of topological spaces. Denote by μ(K) (resp. μr (K) or μn (K)) the cardinal number of different topological types of (resp. reversible or non-reversible) spaces in K. Since each space is reversible or non-reversible, we have μ(K) = μr (K) + μn (K). Recall ([17]) that each subset of any Hausdorff space X with a countable base is homeomorphic to at most c different subsets of X and μ(Ms ) = 2c . Proposition 2.1. (i) μn (Ms ) = 2c and (ii) 2c ≥ μr (Ms ) ≥ c. Proof. Let us prove the equality (i). Note that max{μn (Ms ), μr (Ms )} = 2c and so μr (Ms ) ≤ 2c . If μr (Ms ) < 2c , we have μn (Ms ) = 2c . Assume that μr (Ms ) = 2c . Let F be a family of non-homeomorphic subsets of the Tychonoff cube I ℵ0 . Observe that |F| = 2c . In the space I × I ℵ0 consider the family G = ({{0} × F ) ∪ ({1} × Y ) : F ∈ F} of subsets of I × I ℵ0 , where Y is a subset of I ℵ0 homeomorphic to Nℵ0 . Note that G consists of non-reversible spaces. Moreover, there is a subfamily G  of G which consists of pairwise non-homeomorphic subsets of I × I ℵ0 and |G  | = 2c . Hence μn (Ms ) = 2c . In order to show the inequality (ii) it is enough to observe that there exist families of cardinality continuum consisting of pairwise non-homeomorphic metrizable compacta. So μr (Ms ) ≥ c. 2 We do not know what μr (Ms ) is equal to. In [2] it was proved that μ(Mc ) = c. Moreover, each element of the presented there family of cardinality c consisting of pairwise non-homeomorphic countable subsets of the real line R has a clopen subset homeomorphic to the space Q of rational numbers and hence is non-reversible. Summarizing we get μn(Mc ) = c. Proposition 2.2. c ≥ μr (Mc ) ≥ ℵ1 . Proof. Recall (cf. [12]) that there is a family F of cardinality ℵ1 consisting of pairwise non-homeomorphic infinite countable metrizable compacta. Hence, μr (Mc ) ≥ ℵ1 . 2 We do not know what μr (Mc ) is equal to. But in CH we have evidently μr (Mc ) = μ(Mc ) = ℵ1 = c.

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3. Countable hereditarily reversible Tychonoff spaces of non-countable character Let P ⊆ N∗ = βN \ N. Denote by VP = N ∪ P the subspace of βN. Evidently, χ(VP ) > ℵ0 if P = ∅. Lemma 3.1. Let ∅ = P, Q ⊆ N∗ , f : VP → VQ be a continuous bijection and f (P ) = Q. Then (i) f (N) = N and g = f |N : N → N is a homeomorphism, (ii) βg : βN → βN, βg|N∗ : N∗ → N∗ , βg|VP : VP → βg(VP ) are homeomorphisms, and βg|VP = f , in particular, βg(P ) = Q. Proof. (i) is evident. (ii) Since g is a homeomorphism, βg is also a homeomorphism by a standard argument. Hence, for any A ⊆ βN we have that the mapping βg|A : A → βg(A) is a homeomorphism, in particular, it is valid for A = N∗ and A = VP . Note that f and βg|VP are extensions of g on VP with values in the Hausdorff space βN. Hence they are equal. 2 Remark 3.1. (i) If P, Q ⊆ N∗ , Q finite, |P | ≥ |Q| ≥ 1 and f : VP → VQ is a continuous bijection then f (P ) = Q (in particular, |P | = |Q|). (ii) Note that for each 0 ≤ m < n ≤ ℵ0 we can find subsets P, Q of N∗ with |P | = m, |Q| = n such that there exists a continuous bijection f : VP → VQ which is no homeomorphism. In fact, let m = 0, n = 1. Consider any q ∈ N∗ and set Q = {q}. Define f : V∅ = N → VQ as follows. f (1) = q and f (i + 1) = i for each i ≥ 1. It is clear that f satisfies the required conditions. Let 1 < n < ∞. Consider a decomposition of N into n disjoint infinite subsets Ni , i = 1, . . . , n. Let us note that βN is the disjoint union of sets ClβN Ni , i ≤ n, N∗ is the disjoint union of Bd βN Ni , i ≤ n, each subspace ClβN Ni of βN is homeomorphic to βN, where Ni corresponds to N and Bd βN Ni to N∗ . With the help of the considered case m = 0, n = 1 and the decomposition above one can find P, Q and f with the required properties. Proposition 3.1. For each finite P ⊆ N∗ the space VP is hereditarily reversible. Proof. If P = ∅ the statement is trivial. Let P = ∅ and f : VP → VP be a continuous bijection. Note that f (P ) = P . It follows from Lemma 3.1 that f is a homeomorphism. Hence VP is reversible. Let X be a subspace of VP . It is easy to see that X is homeomorphic to either a copy of Dτ , where τ ≤ ℵ0 or a copy of VQ for some Q ⊆ N∗ with |Q| ≤ |P |. Hence, X is reversible and thus VP is hereditarily reversible. 2 Corollary 3.1. For each x ∈ N∗ the topological union V{x} ⊕ Dℵ0 is hereditarily reversible. Remark 3.2. There exists an infinite countable subset P of N∗ such that VP is not reversible (see the proofs of Propositions 4.3–4.4). By the use of Lemma 3.1 it is easy to see that for every finite subsets P, Q of N∗ such that |P | = |Q| the spaces VP and VQ are not homeomorphic. Let us note that even if |P | = |Q|, the spaces VP and VQ do not need to be homeomorphic. In fact, recall that a space X is homogeneous if for any two points x, y ∈ X there is a homeomorphism h from X onto itself such that h(x) = y. It is well known (see [16] under CH and [10] in ZFC) that the remainder N∗ is not homogeneous. So there are points p, q ∈ N∗ such that there is no homeomorphism h from N∗ onto N∗ with h(p) = q.

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Proposition 3.2. Let p, q ∈ N∗ such that there is no homeomorphism h from N∗ onto N∗ with h(p) = q. Then the spaces V{p} and V{q} cannot be mapped onto each other by continuous bijections (in particular, they are not homeomorphic). Proof. Assume that h : V{p} → V{q} is a continuous bijection. It follows from Lemma 3.1 that there is a homeomorphism r from N∗ onto itself such that r(p) = q. We have a contradiction. Hence, V{p} and V{q} cannot be mapped onto each other by continuous bijections. 2 Recall that a type τ (x, X) of a point x in a space X defines by τ (x, X) = {y ∈ X : there exists a homeomorphism h : X → X with h(x) = y}. In [14] it was proved that there are 2c types of weak P -points in N∗ which are not ω1 -OK points. In particular, there is a subset S of N∗ with cardinality |S| = 2c such that for every x, y ∈ S there is no automorphism h : N∗ → N∗ with h(x) = y. Proposition 3.3. The family VS = {V{x} : x ∈ S} consists of spaces which cannot be mapped onto each other by continuous bijections. Proof. Apply Proposition 3.2. 2 Corollary 3.2. There are 2c different Tychonoff topologies on a countable set such that all but one of its points are clopen sets. Since the cardinal number of all countable subsets of βN ≤ (2c )ℵ0 = 2c , we have the following statement. Corollary 3.3. There are precisely 2c different topological types of countable hereditarily reversible subsets of βN. In particular, if βNc is the class of all countable subsets of βN, then μ(βNc ) = μr (βNc ) = 2c . We do not know what μn (βNc ) is equal to. Problem 3.1. Describe all topological types of countable hereditarily reversible subsets of βN. 4. Uncountable hereditarily reversible Tychonoff spaces of non-countable character and Efimov’s compacta We start with two types of examples of spaces as in the title: non-compact and compact ones. Example 4.1. Let Ω be the space of all ordinal numbers ≤ ω1 , Ω1 the subset of Ω consisting of all countable limit ordinals and Ω2 = Ω \ Ω1 . We will show that each subspace X of Ω such that X ⊆ Ω2 is hereditarily reversible. It is enough to show that X is reversible. In fact, if |X| ≤ ℵ0 or ω1 ∈ / X then X is discrete and hence it is reversible. Let |X| = ℵ1 and ω1 ∈ X. Consider a continuous bijection f : X → X. We will prove that f −1 is continuous. It is evident that f (ω1 ) = ω1 . Since each point p of X \ {ω1 } is clopen in X we need to show that f −1 is continuous at the point ω1 . Consider any open nbd O of ω1 in X. Note that the set A = X \ O has cardinality ≤ ℵ0 . Choose any countable ordinal α in X such that α > β for each β ∈ f (A). It is easy to see that the set V = {γ ∈ X : γ > α} is open in X and f −1 V ⊆ O. Thus f is a homeomorphism. In particular, the subspace Ω2 of the space Ω is hereditarily reversible.

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Let us note that |Ω2 | = ℵ1 , each point of Ω2 except of ω1 is isolated in Ω2 , χ(ω1 , Ω2 ) = ℵ1 and Ω2 is not compact. It is easy to see that the space Ω2 as well as each subspace of Ω2 of cardinality ℵ1 containing the point ω1 is homeomorphic to a space (Y, τ ) with a fixed point p ∈ Y such that the set Y has cardinality ℵ1 and the topology τ is generated by the one-point sets {y}, y ∈ Y \{p}, and the family {Y \A : A ⊆ Y \{p}, |A| ≤ ℵ0 }. Let us note that the construction can be extended to any regular cardinal κ. Example 4.2. One-point compactification cDκ of Dκ , κ > ℵ0 , is hereditarily reversible, |cDκ | = κ and χ(cDκ ) = κ. It is easy to see that cDκ contains a copy of cDℵ0 . The following question seems to be interesting. Problem 4.1. Does there exist compact Hausdorff hereditarily reversible space H of uncountable character different from compacta cDκ , κ > ℵ0 ? We will discuss the problem. Proposition 4.1. Let X be a non-empty compact Hausdorff space without isolated points. Then X contains a subset Y which is homeomorphic to the topological sum ⊕∞ i=1 Xi of countably many infinite compact spaces Xi , i = 1, . . . . Proof. Let p ∈ X. Set V1 = X and choose x1 ∈ V1 \ {p}. Consider an open nbd V2 of p such that ClV2 ⊆ V1 \ {x1 }. Choose x2 ∈ V2 \ {p}. Consider an open nbd V3 of p such that ClV3 ⊆ V2 \ {x2 }. Choose x3 ∈ V3 \ {p} and so on. Enlarge each point xi to an open nbd Oi of xi such that ClOi ⊆ Vi \ ClVi+1 . Then set Xi = ClOi . 2 Let us denote by X  the set of limit points of a space X and by Xiso = X \ X  (i.e. the set of isolated points of X). Proposition 4.2. For each infinite compact Hausdorff space X we have at least one of the following conditions. (a) X is homeomorphic to cDτ , where τ = |X|. (b) X contains a copy of Nℵ0 . (c) X contains a non-empty closed subspace Y without isolated points. In particular, X always contains a copy of Dℵ0 . Proof. Case 1. 0 ≤ |Xiso | < ℵ0 . Set Y = X \ Xiso . Case 2. |Xiso | ≥ ℵ0 . Here we have three subcases. Subcase 2.1. |X  | = 1. Then X is homeomorphic to cDτ , where τ = |X|. Subcase 2.2. 1 < |X  | < ℵ0 . Then X contains a copy of Nℵ0 . Subcase 2.3. |X  | ≥ ℵ0 . Here we have two subcases. Subcase 2.3.1. 0 ≤ |(X  )iso | < ℵ0 . Set Y = X  \ (X  )iso . Subcase 2.3.2. |(X  )iso | ≥ ℵ0 . Then X contains a copy of Nℵ0 . The existence of a copy of Dℵ0 in X follows the structures of the spaces cDκ , κ ≥ ℵ0 , Nℵ0 and Proposition 4.1. 2 Corollary 4.1. Let X be an infinite compact Hausdorff space which is not homeomorphic to cD|X| . If X contains a copy of cDℵ0 then X contains a copy of Nℵ0 .

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Proof. Let p be a limit point for a copy of cDℵ0 in X. If for each open nbd W of p we have |X \ W | < ℵ0 then X is homeomorphic to cD|X| . This is a contradiction. So there exists an open nbd W of p such that |X \ W | ≥ ℵ0 . By Proposition 4.2 the subspace X \ W of X contains a copy of Dℵ0 . So X contains a copy of Nℵ0 . 2 Thus our hypothetical compact space H cannot contain non-trivial converging sequences. It is well known (cf. [5]) that there are compact Hausdorff spaces without non-trivial converging sequences. For example, βN is such a space. We will show that the space βN is not hereditarily reversible. Proposition 4.3. Let X be any non-discrete topological space of cardinality κ ≥ 1 (not necessary even T1 ). If κ ≥ ℵ0 (resp. κ < ℵ0 ), then the space U = (X × Dℵ0 ) ⊕ Dκ (resp. V = (X × Dℵ0 ) ⊕ Dℵ0 ) is not reversible. In particular, if a topological space Z contains a copy of either U or V then Z is not hereditarily reversible. Proof. We will show only the case κ ≥ ℵ0 . Note that the space U is homeomorphic to the topological ∞ sum Z = (⊕∞ i=1 Xi ) ⊕ (⊕i=1 (Dκ )i ), where Xi (resp. (Dκ )i ) is a copy of X (resp. Dκ ) for each integer i ≥ 1. Let f : (Dκ )1 → X1 be any continuous bijection from (Dκ )1 onto X1 , fi,i+1 : Xi → Xi+1 (resp. gi,i+1 : (Dκ )i+1 → (Dκ )i ) be any homeomorphism from Xi onto Xi+1 (resp. (Dκ )i+1 onto (Dκ )i ). Then the mapping F : Z → Z composed from the mappings f, fi,i+1 and gi,i+1 , i ≥ 1, is a continuous bijection which is no homeomorphism. 2 Corollary 4.2. Let X be an infinite compact Hausdorff space. Then X × Dℵ0 is not hereditarily reversible. Proof. Consider an infinite countable subset Y of X. If Y is closed in X, then Y contains a copy of Dℵ0 , and hence X × Dℵ0 contains a copy of Nℵ0 . Otherwise, let Z = Y ∪ {p}, where p ∈ ClX Y . Note that Z is not discrete, |Z| = ℵ0 and Z × Dℵ0 contains a copy of (Z × Dℵ0 ) ⊕ Dℵ0 . By Proposition 4.3 the space (Z × Dℵ0 ) ⊕ Dℵ0 is not reversible. Hence, X × Dℵ0 is not hereditarily reversible. 2 Proposition 4.4. βN is not hereditarily reversible. Proof. In fact, decompose N into countably many infinite disjoint subsets Ni , i = 1, 2 . . . . Note that each subspace ClβN Ni of βN is homeomorphic to βN and it is a clopen subset of βN. Moreover, the sets ClβN Ni , i = 1, 2, . . . are pairwise disjoint. Hence, the subspace ∪∞ i=1 ClβN Ni of βN is homeomorphic to the product βN ×N. By Corollary 4.2 we get that βN is not hereditarily reversible. 2 In [15] the authors stated that βN × N is non-reversible. We have an opposite opinion. Proposition 4.5. βN × N is reversible. Proof. Let us consider a continuous bijection f : βN × N → βN × N. Define Bij = f (βN × {i}) ∩ (βN × {j}), Aij = f −1 Bij , Ji = {j : Bij = ∅} and Ij = {i : Bij = ∅}. It is evident that βN × {i} = ∪{Aij : j ∈ Ji } and βN × {j} = ∪{Bij : i ∈ Ij }. Since f is continuous we have |Ji | < ∞ for each i ≥ 1. Hence every set Aij is clopen in βN × {i}. Let us also note that Aij is homeomorphic to Bij and Bij = f (Aij ). We will show that the set Bij is clopen in βN × {j}. This will imply that f is a homeomorphism. First, let us observe some simple facts about βN that (a) a topological sum X of finitely many spaces Xi , which are either a one-point-space or a copy of βN (there is at least one of such copy in the sum), is homeomorphic to βN;

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(b) any open subset O of βN such that O ∩ N∗ = ∅ contains a copy of βN, in particular, the subspace N∗ does not have isolated points; (c) each clopen subset of βN is either homeomorphic to βN or is a finite subset of N. Since Bij is homeomorphic to Aij and Aij is clopen in βN ×{i}, by (c) each set Bij is either homeomorphic to βN or is a finite subset of βN × {j}. It is enough to show that |Ij | < ∞. In fact, assume that |Ij | = ∞. There are some possibilities which can be reduced (by (a)) to the following situations. (1) βN is a disjoint union K ∪ C of a copy K of βN and an infinite countable set C. (2) βN is a disjoint union ∪∞ n=1 Kn of countably many copies Kn of βN. We will show that both of them are impossible. In fact, (1): Since C is countable, we have (by (b)) that N∗ ⊆ K and |K ∩ N| = ℵ0 . So C ⊆ N and N \ C = K ∩ N. Further, βN = (ClβN C) ∪ (ClβN (K ∩ N)), (ClβN C) ∩ (ClβN (K ∩ N)) = ∅ and K is a disjoint union of ClβN (K ∩ N) and (ClβN C) ∩ N∗ . Since the set (ClβN C) ∩ N∗ is not empty and clopen in K (and in N∗ as well) we get by (b) and (c) that K is not homeomorphic to βN. This is a contradiction. (2): (by a hint of van Mill) There exists an infinite subset L = {ln : n = 1, 2, . . . } of N such that ln ∈ / ∪np=1 Kp for each n. Note that the set ClβN L is clopen in βN and ClβN L is homeomorphic to βN. Let R = ClβN L \ L. It is clear that R is clopen in N∗ and R corresponds N∗ (resp. L corresponds N) under the homeomorphism. By the Baire theorem there is a number m such that O = IntR (Km ∩ R) = ∅. We will show that this leads to an isolated point inside R (and hence in N∗ ) what is impossible (see (b)). In fact, put F = Km ∩ R, and consider a p ∈ O. Let p be an isolated point of Km . So p is an isolated point of F . Observe that there is an open set O of R such {p} = O ∩ F , and O ∩ O = O ∩ O ∩ F = {p}. Since the set O ∩ O is open in R, p is an isolated point of R. Let p be no isolated point of Km . Note that V = (R ∪ (L \ {l1 , . . . , lm }) is open in βN, and V ∩ Km = R ∩ Km = F . Let O be an open set of βN such that O = O ∩ R. Note that the set (O ∩ V ) is open in βN and O ⊆ (O ∩ V ) ∩ Km = O ∩ F ⊆ O ∩ R = O. So (O ∩ V ) ∩ Km = O is an open subset of Km . Since each open subset of Km has an isolated point of Km , the set O contains an isolated point of Km , and hence (by the previous case) there is an isolated point inside R. 2 Question 4.1. Let K be the class of all subsets of βN. What are the cardinals μ(K), μn (K) and μr (K) equal to? Remark 4.1. An inspection of the proof of Proposition 4.3 leads to the following observations: (a) If X = (⊕∞ i=1 Xi ) ⊕Dℵ0 is countable, X1 is non-discrete, and for each i there exists a continuous bijection fi : Xi → Xi+1 of Xi onto Xi+1 , then X is not reversible. (b) If Y = ⊕∞ i=1 Yi is countable, Y1 is non-discrete, and for each i there exists a continuous bijection fi : Yi → Yi+1 of Yi onto Yi+1 , then Y is not hereditarily reversible. Let us summarize that our hypothetical infinite compact Hausdorff hereditarily reversible space H at least may not (a) contain a copy of cDℵ0 , (b) contain a copy of Y from Remark 4.1 (b), which can be found, for example, inside of Z × Dℵ0 , where Z is any infinite compact Hausdorff space (in particular, contain a copy of βN). We have reservations to the existence of such spaces H at least in ZFC.

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In fact, there is an old question by Efimov ([8]): • Does each infinite compact Hausdorff space contain either a copy of βN or a countable converging sequence? In literature infinite compact spaces without copies of βN and countable converging sequences are called Efimov’s compacta. In [9] Fedorchuk under CH constructed for each integer n ≥ 1 an n-dimensional compact Hausdorff space Fn such that all its inifinite closed subsets are n-dimensional. The compacta Fn , n ≥ 1, were first examples of Efimov’s compacta. It is unknown if there is an Efimov’s compactum in ZFC. But there exist Efimov’s compacta under assumptions of different axioms (cf. [11] and [6]). 5. Examples of (non-)reversible and (non-)hereditarily reversible T0 − but not T1 -spaces All considered below spaces are T0 but not T1 . We continue with simple constructions of (non-)reversible spaces. Proposition 5.1. Let X be a connected finite space and κ a cardinal number ≥ 1. Then the space Y = X ×Dκ is reversible. Proof. Let f : Y → Y be a continuous bijection from Y onto itself. It is easy to see that for each d ∈ Dκ there exists t(d) ∈ Dκ such that f (X × {d}) ⊆ X × {t(d)}. Since X is finite and f |X×{d} : X × {d} → X × {t(d)} is a continuous bijection, the mapping f |X×{d} is a homeomorphism. Hence, f is a homeomorphism. 2 Corollary 5.1. Let X and Y be connected finite spaces such that |X| > |Y |, κ be a cardinal number ≥ 1 and n a positive integer. Then the space Z = (X × Dκ ) ⊕ (Y × Dn ) is reversible. Proof. Let f : Z → Z be a continuous bijection of Z onto itself. Since |X| > |Y |, we have that f (X × Dκ ) = X × Dκ and f (Y × Dn ) = Y × Dn . By Proposition 5.1 we have fX×Dκ and fY ×Dn are homeomorphisms. Hence, f is a homeomorphism. 2 Let X be a topological space and p ∈ X. If χ(p, X) = 1 at p, denote by Op the minimal open nbd of the point. Recall that a space X is called Alexandroff if χ(p, X) = 1 for each p ∈ X. Recall also that a space X is called locally finite if each point p of X possesses a finite open nbd. It is easy to see that each locally finite space is Alexandroff, and there are Alexandroff spaces which are not locally finite. For each locally finite space X define Xk = {p ∈ X : |Op | = k}, k = 1, 2, . . . . Lemma 5.1. Let X be a locally finite space, and f : X → X be a continuous bijection from X onto itself. Then f is a homeomorphism iff |f (Op )| = |Of (p) | for each p ∈ X iff f (Xk ) = Xk for each integer k ≥ 1. Proof. We will show the first equivalence. Consider p ∈ X. Since f is continuous, we have f (Op ) ⊆ Of (p) and hence |Op | ≤ |Of (p) |. If f is a homeomorphism, then f (Op ) = Of (p) and hence, |Op | = |Of (p) |. If |Op | = |Of (p) | then f (Op ) = Of (p) . So f is a homeomorphism. 2

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Theorem 5.1. Let X be a locally finite space. (i) Assume that there exists an integer n ≥ 2 such that for each p ∈ X we have |Op | ≤ n, and for each 1 < k ≤ n the set Xk is finite. Then the space X is hereditarily reversible. (ii) Assume that there exists an integer n ≥ 2 such that for each p ∈ X we have |Op | ≤ n, and for each 1 ≤ k < n the set Xk is finite. Then the space X is reversible. (There are both hereditarily reversible and not hereditarily reversible spaces of the type, see Example 5.2.) (iii) Assume that there exists an integer n ≥ 3 such that for each p ∈ X we have |Op | ≤ n, and there exists an integer 1 < m < n such that for each k = m and 1 ≤ k ≤ n the set Xk is finite. Then the space X is reversible. (There are both hereditarily reversible and not hereditarily reversible spaces of the type, see Example 5.2.) (iv) If the set Xk is finite for each k = 1, 2, . . . , then the space X is reversible. (There are both hereditarily reversible and not hereditarily reversible spaces of the type, see Example 5.4 and 5.5.) Proof. First we will show the reversability for all cases. Let f : X → X be a continuous bijection from X onto itself. So |Op | ≤ |Of (p) | for each p ∈ X. It is easy to see that case (i): Starting with k = n, we easily get that f (Xk ) = Xk for each 1 < k ≤ n and hence even f (X1 ) = X1 . case (ii): Starting with k = 1, we get that f (Xk ) = Xk for each 1 ≤ k < n and hence even f (Xn ) = Xn . case (iii): Starting with k = 1, we get that f (Xk ) = Xk for each 1 ≤ k < m, continue with k = n, we get also that f (Xk ) = Xk for each m < k ≤ n. Then we get that even f (Xm ) = Xm . case (iv): Starting with k = 1, we get that f (Xk ) = Xk for each k = 1, 2, . . . . It follows from Lemma 5.1 that f is a homeomorphism for all these cases. Note that if X satisfies the condition (i) then each subspace Y of X also satisfies the condition and hence Y is reversible. This implies that X is hereditarily reversible in this case. 2 Theorem 5.1(i) and Proposition 4.3 imply the following statement. Corollary 5.2. Let X be a finite space. Then the space X ⊕ Dκ , κ ≥ 1, is hereditarily reversible. Moreover, if X is not discrete then the space Y = (X ⊕ Dℵ0 )2 is not reversible. Proof. Let us only note that the subspace (X × Dℵ0 ) ⊕ (Dℵ0 × Dℵ0 ) of Y is not reversible and it is clopen in Y . 2 Example 5.1. Let C2p be a connected T0 -space consisting of two points, and E = C2p × Dℵ0 . It follows from Proposition 5.1 that the space Y is reversible. Since E contains a copy of the space E ⊕ Dℵ0 (which is not reversible by Proposition 4.3), the space E is not hereditarily reversible. Let us notice that E is a locally finite T0 -space (no T1 -space) such that |E1 | = |E2 | = ℵ0 and |Ei | = 0 for each i ≥ 3, where Ei = {p ∈ E : |Op | = i}. It follows from Corollary 5.2 (resp. Corollary 5.1) that the space (C2p × Dk ) ⊕ Dℵ0 (resp. E ⊕ Dk ), which can be embedded in E, is reversible for each positive integer k. This implies that each non-reversible subspace of E is homeomorphic to the space E ⊕ Dℵ0 . Remark 5.1. It is easy to observe that each non-discrete Alexandroff T0-space X contains a copy of the space C2p . Hence, the product X × Dκ , where κ ≥ ℵ0 , contains a copy of the space E, and thus X × Dκ is not hereditarily reversible.

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Question 5.1. Let X be a locally finite T0 -space. Assume that there exists an integer n ≥ 2 such that for each p ∈ X we have |Op | ≤ n, and there exist also integers 1 ≤ k < l ≤ n such that the sets Xk and Xl are infinite. Is the space X not hereditarily reversible? (For k = 1 and l = 2 the answer is positive, see Proposition 5.3.) Let X be an Alexandroff T0 -space and A ⊆ X. We put stA = ∪{Oq : Oq ∩A = ∅, q ∈ X}. Consider p ∈ X. Put V1,p = Op , A1,p = ClX V1,p , V2,p = stV1,p , A2,p = ClX V2,p , . . . , Vn,p = stVn−1,p , An,p = ClX Vn,p , n ≥ 3 and Σp = ∪∞ i=1 Vi,p . It is easy to see that Vi,p ⊆ Ai,p ⊆ Vi+1,p for each integer i ≥ 1. Lemma 5.2. Let X be an Alexandroff T0 -space and p, q ∈ X. Then the following are valid. (i) If Vn,p \ An−1,p = ∅ for some n ≥ 2 then An−1,p = Vn,p = Am,p = Vm+1,p for each integer m ≥ n. (ii) If Vn,p \ An−1,p = ∅ for each n ≥ 2 then X contains a subspace homeomorphic to the space E. (iii) If Vn,p ∩ Vm,q = ∅ for some positive integers n, m then Σp = Σq . In particular, X is a disjoint union of clopen sets Σp , p ∈ P , where P is a subset of X. Proof. (i) Assume that Vn,p = An−1,p = P . Note that P is a clopen subset of X. Hence P = Am,p = Vm+1,p for each integer m ≥ n. (ii) Let xn ∈ Vn,p \ An−1,p for each n ≥ 2. Note that for each n ≥ 2 there exists yn ∈ An−1,p such that xn ∈ Oyn . Put Fn = {x2n , y2n }, n ≥ 1, and F = ∪∞ i=1 Fn . Note that F1 = F ∩ V2,p , Fn = F ∩ V2n,p \ A2(n−1),p for each n ≥ 2 and all Fn , n ≥ 1, are disjoint. Since for each n ≥ 1 the subspace Fn of X is homeomorphic to the space C2p , the subspace F of X is homeomorphic to the space E. (iii) It is evident. 2 Theorem 5.2. Let X be an infinite Alexandroff T0 -space with |stOp | < ℵ0 for each point p ∈ X and X be not homeomorphic to any space F ⊕ Dκ , where κ ≥ ℵ0 , and F is a finite space. Then X contains a subspace homeomorphic to the space E. In particular, X is not hereditarily reversible. Proof. There are two possibilities. (a) There exists a point p ∈ X such that Vn,p \ An−1,p = ∅ for each n ≥ 2. It follows from Lemma 5.2 (ii) that we can find a subspace of X which is homeomorphic to the space E. (b) For each p ∈ X there exists an integer np such that Vnp ,p \ Anp −1,p = ∅. It follows from Lemma 5.2 (i) np that Σp = ∪i=1 Vi,p . Note that |Σp | < ℵ0 and Σp is clopen in X (see Lemma 5.2 (iii)). Moreover, there exist countably many points pn , n ≥ 1, such that the sets Σpn , n ≥ 1, are disjoint and |Σpn | > 1 for each n. The last implies that each set Σpn contains the space C2p . Now it is easy to see that X contains a copy of the space E. 2 Let us denote by A∗ℵ0 the class of Alexandroff T0 -spaces X with |stOp| < ℵ0 for each point p ∈ X. It is easy to see that the Khalimsky line K (cf. [1]) is an element of A∗ℵ0 as well as other examples of digital topology. Theorem 5.2 implies the following characterization of hereditarily reversible spaces in A∗ℵ0 . Corollary 5.3. The only hereditarily reversible elements of A∗ℵ0 are the sums F ⊕ Dκ , where κ ≥ 0 and F are finite spaces. It is easy to see that each space from A∗ℵ0 is locally finite. Let us indicate some simple facts about the class A∗ℵ0 .

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Proposition 5.2. (i) (ii) (iii) (iv)

X ∈ A∗ℵ0 iff X is an Alexandroff T0 -space such that |st{p}| < ℵ0 for each point p ∈ X. Let X ∈ A∗ℵ0 and Y ⊆ X then Y ∈ A∗ℵ0 . Let Xγ ∈ A∗ℵ0 , γ ∈ Γ. Then the topological sum ⊕γ∈Γ Xγ also belongs to A∗ℵ0 . Let X, Y ∈ A∗ℵ0 . Then the topological product X × Y also belongs to A∗ℵ0 .

Proof. (i)–(iii) are evident. For (iv) let us only note that for each x ∈ X and y ∈ Y we have O(x,y) = Ox ×Oy , and st{(x, y)} ⊆ st{x} × st{y}. 2 Corollary 5.4. For each integer n ≥ 1 we have Kn ∈ A∗ℵ0 . Example 5.2. Let N be the set of positive integers, and let O1 = {1}, On = {1, n}, n ≥ 2. Endow N with the topology τ1 defined by the base B1 = {On : n ∈ N}. Note that (N, τ1 ) is a locally finite T0 -space such that |O1 | = 1 and |On | = 2 for each n ≥ 2. So we have |N1 | = 1, |N2 | = ℵ0 and |Nk | = 0 for each k ≥ 3, where Nk = {n ∈ N : |On | = k}, k ≥ 1. It follows from Theorem 5.1(ii) that the space (N, τ1 ) is reversible. Since each subspace of (N, τ1 ) is either a finite space, Dℵ0 or a copy of (N, τ1 ), we get that the space (N, τ1 ) is hereditarily reversible. Note that even each space (N, τ1 ) ⊕ Y , where Y is a finite space, is hereditarily reversible (apply Theorem 5.1 (iii) and Corollary 5.2). Let F = (N, τ1 ) ⊕ Dℵ0 and Dℵ0 = {a1 , a2 , . . . }. Consider a mapping f : F → F such that f (a1 ) = 2, f (1) = 1, f (ai+1 ) = ai for each i ≥ 1 and f (i) = i + 1 for each i ≥ 2. Note that f is a continuous bijection, however f −1 is not continuous. So F is not reversible. Similarly, it follows from Theorem 5.1(ii) that the space (N, τ1 ) × Dk is reversible for each integer k ≥ 2. Since each (N, τ1 ) × Dk contains a subspace homeomorphic to F , it is not hereditarily reversible. / A∗ℵ0 . Let us also observe that the construction can be applied to any Note that |stO1 | = ℵ0 so (N, τ1 ) ∈ set S with cardinality ≥ 2 and a fixed point s ∈ S (instead of N with the point 1). We will denote the space by (S, τ1 ). It is easy to see that if |S| = 2 then (S, τ1 ) is homeomorphic to the space C2p . Proposition 5.3. Let X be a locally finite T0 -space such that X1 and X2 are infinite, where Xi = {x ∈ X : |Ox| = i}. Then the space X is not hereditarily reversible. Proof. Put Y = X1 ∪ X2 . For each q ∈ X2 there exists p(q) ∈ X1 such that Oq = {q, p(q)}. Define P = {p(q) : q ∈ X2 } ⊆ X1 . If |P | ≥ ℵ0 then it is easy to see that Y contains a copy of E and so Y (and hence X) is not hereditarily reversible. If |P | < ℵ0 then the set X1 \ P is clopen, infinite and discrete in Y , and the set X2 ∪ P contains a copy of the space (N, τ1 ). From the facts it follows that Y contains a copy of F . So Y (and hence X) is not hereditarily reversible. 2 Let k be an integer ≥ 2. Denote by Ak the class of locally finite T0 -spaces X such that Xm = ∅ for each m ≥ k. It is evident that each class Ak is monotone w r t subsets. Note that F, E, (S, τ1 ) × Dκ ∈ A2 . Theorem 5.3. The only hereditarily reversible elements of A2 are finite elements of A2 , the topological sums of one infinite discrete space and one finite element of A2 , the topological sums of one space of type (S, τ1 ) with |S| ≥ ℵ0 and one finite element of A2 . (Note that each finite element of A2 is the topological sum of one finite discrete space and finitely many spaces of type (S, τ1 ) with |S| < ℵ0 .)

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Proof. Let X ∈ A2 . Hence, X = X1 ∪ X2 . There are four possibilities. Case 1: X1 and X2 are finite. So X is finite and hence it is hereditarily reversible. Case 2: X1 is infinite and X2 is finite. It follows from Theorem 5.1 that X is hereditarily reversible. It is easy to see that X is the topological sum of Dκ , where κ ≥ ℵ0 , and a finite element of A2 . Case 3: X1 is finite and X2 is infinite. By the use of Example 5.2 it is easy to see that the only hereditarily reversible spaces X are the topological sums of one space of type (S, τ1 ) with |S| ≥ ℵ0 and one finite element of A2 . Case 4: X1 and X2 are infinite. It follows from Proposition 5.3 that such X is not hereditarily reversible. 2 Corollary 5.5. Let X ∈ A2 . Then X is not hereditarily reversible iff X contains either E or F . Problem 5.1. Let k be an integer ≥ 3. Characterize the hereditarily reversible spaces in the class Ak . Example 5.3. Let V = {(i, j) : 1 ≤ i ≤ j, j = 1, 2, . . . }, and let O(i,j) = {(k, j) : 1 ≤ k ≤ i} for each (i, j) ∈ V. Endow V with the topology τ2 defined by the base B2 = {Ox : x ∈ V}. Note that (V, τ2 ) ∈ A∗ℵ0 \ Ak for each integer k. It is easy to see that (V, τ2 ) is reversible and contains E. Hence it is not hereditarily reversible (one can use also Corollary 5.3). Let us recall that if X is an Alexandroff T0 -space then for every two points p, q ∈ X we have the following: p ∈ Oq iff Op ⊆ Oq . The equivalence defines a partial order “≤” on X: p ≤ q iff Op ⊆ Oq . We will call a point p ∈ X a maxpoint of X if there is no other point q ∈ X such that p ≤ q. The set of maxpoints of X we will denote by MX . Let Amax be the class of locally finite T0 -spaces X such that MX = ∅ and for each point p ∈ X we have ℵ0 p ≤ q for some q ∈ MX . and An ⊆ Amax for each n ≥ 1. It is easy to see that A∗ℵ0 ⊆ Amax ℵ0 ℵ0 Example 5.4. Let N be the set of positive integers, and let On = {m ∈ N : m ≤ n}, n ∈ N. Endow N with the topology τ3 defined by the base B3 = {On : n ∈ N}. Note that (N, τ3 ) is a locally finite T0 -space such that |On | = n for each n ∈ N. So we have |Nk | = 1 for each k ≥ 1, where Nk = {n ∈ N : |On | = k}, k ≥ 1. It follows from Theorem 5.1(iii) that the space (N, τ3 ) is reversible. Since each subspace of (N, τ3 ) is either a finite space or a copy of (N, τ3 ), we get that the space (N, τ3 ) is hereditarily reversible. Let G = (N, τ3 ) ⊕ Dℵ0 and Dℵ0 = {a1 , a2 , . . . }. Consider a mapping g : G → G such that f (a1 ) = 1, f (ai+1 ) = ai and f (i) = i + 1 for each i ≥ 1. Note that g is a continuous bijection, however g −1 is not continuous. So G is not reversible. Similarly, it follows from Theorem 5.1(iv) that the space (N, τ3 ) × Dk is reversible for each integer k ≥ 2. Since each subspace of (N, τ3 ) × Dk is either a finite space, (N, τ3 ) × Dl or X ⊕ ((N, τ3 ) × Dl ), where X is a finite space and l is an integer ≤ k, we get that the space (N, τ3 ) × Dk is hereditarily reversible. Note that G, (N, τ3 ) ∈ / Amax ℵ0 . Example 5.5. Let T = {0} ∪ {(0, i1 , . . . , ik ) : i ∈ {0, 1}, k ≥ 1}. Let O0 = {0} and O(0,i1 ,...,ik ) = {0, (0, i1 ), . . . , (0, i1 , . . . , ik )} for each point (0, i1 , . . . , ik ) ∈ T. Endow T with the topology τ4 defined by the base B4 = {Op : p ∈ T}. Note that (T, τ4 ) is a locally finite T0 -space such that |O0 | = 1 and |O(0,i1 ,...,ik ) | = k + 1 for each (0, i1 , . . . , ik ) ∈ T. So we have |Tk | = 2k for each k ≥ 1, where Tk = {p ∈ T : |Op | = k}, k ≥ 1. It follows from Theorem 5.1(iii) that the space (T, τ4 ) is reversible. It is easy to see that the space (T, τ4 ) contains a subspace homeomorphic to (T, τ4 ) × Dℵ0 and hence even a subspace homeomorphic to the space G. Thus the space (T, τ4 ) is not hereditarily reversible. Note that (T, τ4 ) ∈ / Amax ℵ0 .

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Problem 5.2. Characterize the hereditarily reversible spaces in the class Amax ℵ0 . Problem 5.3. Characterize the hereditarily reversible spaces in the class of all locally finite T0 -spaces. References [1] C. Adams, R. Franzosa, Introduction in Topology Pure and Applied, Pearson, 2009. [2] W. Brian, J. van Mill, R. Suabedissen, Homogeneity and generalizations of 2-point sets, Houst. J. Math. 40 (3) (2013) 885–898. [3] V.A. Chatyrko, Y. Hattori, On reversible and bijectively related topological spaces, Topol. Appl. 201 (2016) 432–440. [4] V.A. Chatyrko, A. Karassev, Reversible Spaces and Products, Topol. Proc. 49 (2017) 321–325. [5] A. Dow, K.P. Hart, Cech–Stone compactifications of N and R, in: Encyclopedia of General Topology, Elsevier, NorthHolland, 2004. [6] A. Dow, S. Shelah, An Efimov space from Martin’s axiom, Houst. J. Math. 39 (4) (2013) 1423–1435. [7] P.H. Doyle, J.G. Hocking, Bijectively related spaces I: manifolds, Pac. J. Math. 11 (1984) 23–33. [8] B.A. Efimov, The embedding of the Stone–Cech compactification of discrete spaces into becompacta, Dokl. Akad. Nauk SSSR 189 (1969) 244–246 (in Russian). [9] V.V. Fedorchuk, A bicompactum whose infinite closed subsets are all n-dimensional, Math. Sb. 96 (138) (1975) 41–62. [10] Z. Frolik, Non-homogeneity of βP \ P , Comment. Math. Univ. Carol. 8 (1967) 705–709. [11] K.P. Hart, Efimov’s problem, in: E. Peral (Ed.), Open Problems in Topology II, Elsevier Science, 2007, pp. 171–177. [12] H. Kato, Jong-Jin Park, Expansive homeomorphisms of countable compacta, Topol. Appl. 95 (1999) 207–216. [13] J. Kulesza, Some results on spaces between the Sorgenfrey topology and the usual topology on the reals, manuscript, 2015. [14] J. van Mill, Types of weak P -points in βN \ N, in: General Topology and Its Relation to Modern Analysis and Algebra, V, Prague, 1981, pp. 481–485. [15] M. Rajagopalan, A. Wilansky, Reversible topological spaces, J. Aust. Math. Soc. 61 (1966) 129–138. [16] W. Rudin, Homogeneity problems in the theory of Čech compactifications, Duke Math. J. 23 (1956) 409–419. [17] W. Sierpinski, General Topology, N.Y. Dover Publications, Mineola, 2000.