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Subspaces (Hereditary (P)-Spaces)
b-1
b-1
Subspaces (hereditary (P)-spaces)
31
Subspaces (Hereditary (P)-Spaces)
Let X be a given topological space with a topology O (the collection of open subsets) and X a subset of X. Put O = X ∩ O: O ∈ O . Then O satisfies the conditions to be a topology on X ; that is, X is a topological space with the topology O . This topological space X is called a subspace of X and this topology O is called the relative topology (the induced topology, the subspace topology) of O with respect to X . Sometimes, a closed subset (open subset) is called a closed subspace (open subspace) in view of subspaces. The following facts are direct consequences of definitions of subspaces. (1) A subset E of X is a closed (open) subset of the space X if and only if E = X ∩ E for some closed (open) subset E of the space X. (2) Let A be a subset of X and A¯ denote the closure of A in X . Then A¯ = X ∩ A¯ holds, where A¯ denotes the closure of A in X. (3) For a point x of X a subset U of X is a neighbourhood of x in X if and only if U = X ∩ U for some neighbourhood U of x in X. (4) For a filter F in X is convergent in X if and only if as a filter in X, F converges to a point of X . (5) For a net N in X , N is convergent in X if and only if as a net in X, N converges to a point of X . In particular, for a sequence S in X , S is convergent in X if and only if S converges to a point of X . (6) If B is a base for X then B = {B ∩ X : B ∈ B} is a base for X . There is a notion related to that of a base that is occasionally useful when dealing with subspaces: an outer base for a subspace X of X is a family B of open sets in X such that for every point y of X and every open set O in X with x ∈ O there is a B ∈ B with x ∈ B ⊆ O. If B is a base for X then {B ∈ B: B ∩ X = ∅} is an outer base for X . Let X be the Euclidean plane R2 and let X be the x-axis of R2 . Then X is a closed subspace of X. More generally, for any two natural numbers m and n with m n, the m-dimensional Euclidean space Rm is a closed subspace of the n-dimensional Euclidean space Rn . Similarly, subspaces are defined for uniform spaces and proximity spaces; that is, the restriction of the uniformity or the proximity relation to a subset forms a uniformity or a proximity relation on the subspace, respectively. Let X be a topological space and Y a subspace of X. If Y = X holds, then Y is called a dense subspace of X. For the real line R1 the subspace consisting of all rational numbers is a dense subspace of R1 and the subspace consisting of all irrational numbers is also a dense subspace of R1 .
On the other hand, it is easily seen that R1 is not a dense subspace of Rn for any n 2. For a topological property P we say that P is a hereditary property (closed-hereditary property or a openhereditary property) if any subspace (closed subspace or open subspace) of any topological space possessing property P also has the property P. Any hereditary property is always a closed hereditary property and an open hereditary property, as well. Converses of above directions are very often not true. The following properties are hereditary properties which are easily seen from their definitions. (1) (2) (3) (4) (5) (6)
T0 , T1 , T2 , regularity, complete regularity, metrizability.
The following properties are closed hereditary ones but not hereditary ones. (1) (2) (3) (4) (5) (6)
normality, compactness, paracompactness, Lindelöf property, local compactness, completeness.
To show that above properties (1), (2), (3) and (4) are not hereditary the following space is useful. Tychonoff plank Let X be the product space [0, ω1 ] × [0, ω0 ] and Y = X − {(ω1 , ω0 )}. Then X is a compact T2 space and Y is an open subspace of X. Y is sometimes called the Tychonoff plank [E, 3.12.20]. Since X is a compact T2 space, X satisfies properties (1), (2), (3) and (4). On the other hand, Y is not normal, because the two edges {ω1 } × [0, ω0 ) and [0, ω1 ) × {ω0 } are not separated by disjoint open subsets of Y . Since Y is not normal, Y does not satisfy any of the properties (2), (3) and (4). To show that the local compactness is not hereditary, put X = R1 (real line) and Y = Q (the set of all rational numbers) as a subspace of X. Then X is a locally compact space, but Y is not locally compact subspace of X. To show that the completeness is not hereditary, put X = [0, 1] (the unit interval as a subspace of R1 ) and Y = [0, 1) as a subspace of X. Then X is a complete metric space with respect to the usual metric, but Y is not a complete subspace, because any sequence in Y , convergent to 1 is a Cauchy sequence in Y which does not converge in Y . In the case of
32 completeness it is to be noted that the Cauchyness depends on the metric. The following properties are open hereditary but not hereditary ones. (1) local compactness, (2) local connectedness, (3) separability. To show that above properties (1) and (2) are not hereditary, put X = R1 and Y = Q as a subspace of X. Then X satisfies properties (1) and (2), but Y does not satisfy either of the properties (1) or (2). To show that the property (3) is not hereditary, the following space is used as a representative example. Niemytzki plane Let X = {(x, y) ∈ R2 : y 0} be the subset of R2 . As a base for open subsets at a point (x, y) for which y > 0, we choose the interiors of circles around (x, y), and for a point (x, 0) on x-axis, we choose as a base sets consisting of the point (x, 0) and the interior of a circle tangent to the x-axis at the point (x, 0). The space X is called the Niemytzki Plane [E, 1.2.4] or the Niemytzki space [N, Example II.5, Example III.3]. Then X is separable, because the set of points, both of whose coordinates are rational numbers is a countable dense subset of X. On the other hand, Y is not a separable subspace, because Y is an uncountable discrete space. There are many topological properties which are not closed hereditary and not open hereditary, as well. For example, connectedness is such a property. To see this take R1 and its subspaces [0, 1] ∪ [2, 3] and (0, 1) ∪ (2, 3). Then R1 is a connected space, and the first subspace is closed and the second one is an open subspace, but neither of them are connected. For a hereditary property P, a space with the property P is called a hereditary (P)-space [N, p. 102]. A most familiar such property is metrizability; that is, any subspace of any metrizable space is metrizable. For a property (P) which is not necessarily hereditary in general, we can classify spaces according to whether the
Section B:
Basic constructions
property (P) hold hereditarily or not. For example, although the separability is not a hereditary property, the real line R1 is a hereditarily separable space. More generally, a new space is defined as the hereditarily (P)-space; for instance, a hereditarily normal space means the space in which any subspace is normal. Generally speaking, there are two types of topological properties. The first one is the property which characterizes spaces themselves; for instance, compactness, paracompactness and normality etc. are conditions concerning for whole spaces, and the second one is the property which is characterized using subsets; for instance, separability and nowhere density etc. are characterized by the conditions for subsets. For every topological space X and every subspace Y of X the formula iY (x) = x defines a map iY of Y into X. The map iY is called the embedding of Y in X. Let X and X1 be two topological spaces. If for a subspace Y of X there exists a homeomorphism f : X1 → Y , then we call the space X1 embeddable in X and the map iY f : X1 → X a homeomorphic embedding of X1 in X. Sometimes, X is called an extension of X1 [E, p. 67]. Let P be a topological property and X a topological space. If for every point x of X there exists a neighbourhood Vx of x which has the property P, then we say that X has property P locally. A subset A of X is called a locally closed subset, if A is closed locally. This means that every point x of A has a neighbourhood such that Vx ∩ A is a closed subset of Vx . In other words, A is closed locally if and only if A = O ∩ F for some open subset O of X and some closed subset F of X. As an example of a locally closed subset, let A = [0, 1) be the right open interval as the subset of R1 . Then A is a locally closed subset of R1 ; in other word, A is the intersection of a closed interval [0, 1] and an open interval (−1, 1) [E, 2.7.1].
Akihiro Okuyama Kobe, Japan
b-1
Subspaces (hereditary (P)-spaces)
31
Subspaces (Hereditary (P)-Spaces)
Let X be a given topological space with a topology O (the collection of open subsets) and X a subset of X. Put O = X ∩ O: O ∈ O . Then O satisfies the conditions to be a topology on X ; that is, X is a topological space with the topology O . This topological space X is called a subspace of X and this topology O is called the relative topology (the induced topology, the subspace topology) of O with respect to X . Sometimes, a closed subset (open subset) is called a closed subspace (open subspace) in view of subspaces. The following facts are direct consequences of definitions of subspaces. (1) A subset E of X is a closed (open) subset of the space X if and only if E = X ∩ E for some closed (open) subset E of the space X. (2) Let A be a subset of X and A¯ denote the closure of A in X . Then A¯ = X ∩ A¯ holds, where A¯ denotes the closure of A in X. (3) For a point x of X a subset U of X is a neighbourhood of x in X if and only if U = X ∩ U for some neighbourhood U of x in X. (4) For a filter F in X is convergent in X if and only if as a filter in X, F converges to a point of X . (5) For a net N in X , N is convergent in X if and only if as a net in X, N converges to a point of X . In particular, for a sequence S in X , S is convergent in X if and only if S converges to a point of X . (6) If B is a base for X then B = {B ∩ X : B ∈ B} is a base for X . There is a notion related to that of a base that is occasionally useful when dealing with subspaces: an outer base for a subspace X of X is a family B of open sets in X such that for every point y of X and every open set O in X with x ∈ O there is a B ∈ B with x ∈ B ⊆ O. If B is a base for X then {B ∈ B: B ∩ X = ∅} is an outer base for X . Let X be the Euclidean plane R2 and let X be the x-axis of R2 . Then X is a closed subspace of X. More generally, for any two natural numbers m and n with m n, the m-dimensional Euclidean space Rm is a closed subspace of the n-dimensional Euclidean space Rn . Similarly, subspaces are defined for uniform spaces and proximity spaces; that is, the restriction of the uniformity or the proximity relation to a subset forms a uniformity or a proximity relation on the subspace, respectively. Let X be a topological space and Y a subspace of X. If Y = X holds, then Y is called a dense subspace of X. For the real line R1 the subspace consisting of all rational numbers is a dense subspace of R1 and the subspace consisting of all irrational numbers is also a dense subspace of R1 .
On the other hand, it is easily seen that R1 is not a dense subspace of Rn for any n 2. For a topological property P we say that P is a hereditary property (closed-hereditary property or a openhereditary property) if any subspace (closed subspace or open subspace) of any topological space possessing property P also has the property P. Any hereditary property is always a closed hereditary property and an open hereditary property, as well. Converses of above directions are very often not true. The following properties are hereditary properties which are easily seen from their definitions. (1) (2) (3) (4) (5) (6)
T0 , T1 , T2 , regularity, complete regularity, metrizability.
The following properties are closed hereditary ones but not hereditary ones. (1) (2) (3) (4) (5) (6)
normality, compactness, paracompactness, Lindelöf property, local compactness, completeness.
To show that above properties (1), (2), (3) and (4) are not hereditary the following space is useful. Tychonoff plank Let X be the product space [0, ω1 ] × [0, ω0 ] and Y = X − {(ω1 , ω0 )}. Then X is a compact T2 space and Y is an open subspace of X. Y is sometimes called the Tychonoff plank [E, 3.12.20]. Since X is a compact T2 space, X satisfies properties (1), (2), (3) and (4). On the other hand, Y is not normal, because the two edges {ω1 } × [0, ω0 ) and [0, ω1 ) × {ω0 } are not separated by disjoint open subsets of Y . Since Y is not normal, Y does not satisfy any of the properties (2), (3) and (4). To show that the local compactness is not hereditary, put X = R1 (real line) and Y = Q (the set of all rational numbers) as a subspace of X. Then X is a locally compact space, but Y is not locally compact subspace of X. To show that the completeness is not hereditary, put X = [0, 1] (the unit interval as a subspace of R1 ) and Y = [0, 1) as a subspace of X. Then X is a complete metric space with respect to the usual metric, but Y is not a complete subspace, because any sequence in Y , convergent to 1 is a Cauchy sequence in Y which does not converge in Y . In the case of
32 completeness it is to be noted that the Cauchyness depends on the metric. The following properties are open hereditary but not hereditary ones. (1) local compactness, (2) local connectedness, (3) separability. To show that above properties (1) and (2) are not hereditary, put X = R1 and Y = Q as a subspace of X. Then X satisfies properties (1) and (2), but Y does not satisfy either of the properties (1) or (2). To show that the property (3) is not hereditary, the following space is used as a representative example. Niemytzki plane Let X = {(x, y) ∈ R2 : y 0} be the subset of R2 . As a base for open subsets at a point (x, y) for which y > 0, we choose the interiors of circles around (x, y), and for a point (x, 0) on x-axis, we choose as a base sets consisting of the point (x, 0) and the interior of a circle tangent to the x-axis at the point (x, 0). The space X is called the Niemytzki Plane [E, 1.2.4] or the Niemytzki space [N, Example II.5, Example III.3]. Then X is separable, because the set of points, both of whose coordinates are rational numbers is a countable dense subset of X. On the other hand, Y is not a separable subspace, because Y is an uncountable discrete space. There are many topological properties which are not closed hereditary and not open hereditary, as well. For example, connectedness is such a property. To see this take R1 and its subspaces [0, 1] ∪ [2, 3] and (0, 1) ∪ (2, 3). Then R1 is a connected space, and the first subspace is closed and the second one is an open subspace, but neither of them are connected. For a hereditary property P, a space with the property P is called a hereditary (P)-space [N, p. 102]. A most familiar such property is metrizability; that is, any subspace of any metrizable space is metrizable. For a property (P) which is not necessarily hereditary in general, we can classify spaces according to whether the
Section B:
Basic constructions
property (P) hold hereditarily or not. For example, although the separability is not a hereditary property, the real line R1 is a hereditarily separable space. More generally, a new space is defined as the hereditarily (P)-space; for instance, a hereditarily normal space means the space in which any subspace is normal. Generally speaking, there are two types of topological properties. The first one is the property which characterizes spaces themselves; for instance, compactness, paracompactness and normality etc. are conditions concerning for whole spaces, and the second one is the property which is characterized using subsets; for instance, separability and nowhere density etc. are characterized by the conditions for subsets. For every topological space X and every subspace Y of X the formula iY (x) = x defines a map iY of Y into X. The map iY is called the embedding of Y in X. Let X and X1 be two topological spaces. If for a subspace Y of X there exists a homeomorphism f : X1 → Y , then we call the space X1 embeddable in X and the map iY f : X1 → X a homeomorphic embedding of X1 in X. Sometimes, X is called an extension of X1 [E, p. 67]. Let P be a topological property and X a topological space. If for every point x of X there exists a neighbourhood Vx of x which has the property P, then we say that X has property P locally. A subset A of X is called a locally closed subset, if A is closed locally. This means that every point x of A has a neighbourhood such that Vx ∩ A is a closed subset of Vx . In other words, A is closed locally if and only if A = O ∩ F for some open subset O of X and some closed subset F of X. As an example of a locally closed subset, let A = [0, 1) be the right open interval as the subset of R1 . Then A is a locally closed subset of R1 ; in other word, A is the intersection of a closed interval [0, 1] and an open interval (−1, 1) [E, 2.7.1].
Akihiro Okuyama Kobe, Japan