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On non-cut points and COTS. (English) Zbl 1408.54007

A connected ordered topological space (COTS) is a connected space \(X\) in which given any subset \(Y\) consisting of exactly three points, there is \(y\in Y\) such that \(Y\) meets both components of \(X\setminus \{y\}\) (no separation axiom is assumed). A subset \(Y\) of the space \(X\) is said to be cut point convex if whenever \(a,b\in Y\), then the subset \(S(a,b)\) of all points of \(X\) which separate \(X\) between \(a\) and \(b\) lies in \(Y\) also. A subset \(Y\) of \(X\) is said to be an \(H\)-\(set\) if every cover of \(Y\) by open subsets of \(X\) contains a finite family whose union is dense in \(Y\). This paper studies relations between cut point convex sets, \(H\)-sets and COTS. For example, among many other results it is shown that (1) If \(X\) is a COTS and \(H\) is an \(H\)-subset of \(X\), then \(H\subseteq S(a,b)\cup\{a,b\}\) for some \(a,b\in H\); and (2) If \(X\) is a connected space with at most two non-cut points such that every proper, non-degenerate regular closed connected subset \(K\) contains only a finite number of closed points of \(X\), then \(X\) is (homeomorphic to) a finite connected subspace of the Khalimsky line.

MSC:

54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces
54F15 Continua and generalizations
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References:

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