Arc components in metric continua

Topology and its Applications North-Holland

22 (1986) 83-84

ARC COMPONENTS Attilio

IN METRIC

CONTINUA

LE DONNE

Universitci di Padova, Seminario

Matematico,

Istituto di Algebra e Geometria,

35131 Padova, Italy

Received 5 November 1984 Revised 30 January 1985

There is a metric

continuum

AMS (MOS)

Subj. Class.:

1 Arc component

in which every arc component

Primary

Bore1 set

is not a Bore1 set.

54D05, E45, HO5 metric continua

1. Introduction It is well known that the component of any point in any topological space is closed, but that this need not hold for arc components. In [l] Kunen and Starbird show that there is a metric continuum in which some arc component is not a Bore1 set, and they ask for a continuum in which every arc component is not a Bore1 set. In this paper

we construct

2. Outline of the construction

such a space that is a metric continuum.

of a continuum with every arc component not Bore1

We start from the continuum M given in [l]. This space has a point p whose arc component

is not Borel. It also contains

a

closed subset F such that pi F, F = C x N* (where C is the Cantor set and N* is the one-point compactification of N), and F intersects every arc component of M. Let Y-MxC; if A={p}xC and B=FxC we have that AnB=@ A-C, B = C x N* x C, for every point q of A the arc component of q is not Bore1 and every arc component of Y intersects B. Since C x N* x C = C, B is homeomorphic to A. Let Z=(U{Yx{n}: no N})/where we identify Bx{n} with Ax{n+l} for each n E N. Every arc component of Z is not Bore1 but Z is neither compact nor necessarily connected. If we make the one point compactification of Z we obtain a space that 0166-8641/86/$3.50

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1986, Elsevier

Science

Publishers

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A. Le Donne / Arc components in metric continua

84

is arc connected,

so we make a compactification

of Z with the interval

I = [-1,

l]

in such a way that W = Z v I and I is an arc component of W. To do so we consider a continuous map cp from Z to (0, l] such that cp( Y x {n}/-) = [l/n + 1, l/n] and we consider continuous

in W the weakest topology that contains the open sets of Z and makes the map $?: W+ R2 defined by 27?(x) = (0, x) if x E I and S?(z) =

(cp(z),sin(l/cp(z))) if zEZ. The desired continuum is W/Clearly it is a metric space.

where I is identified

with some arc of Z.

Reference [l]

K. Kunen

and M. Starbird,

Arc components

in metric continua,

Topology

Appl. 14 (1982) 167-170.