S(n)-\(\theta\)-closed spaces. (English) Zbl 0658.54015

Dikranjan, D.; Giuli, E.

Topology Appl. 28, No. 1, 59-74 (1988).

For a subset M of a topological space X, the \(\theta\)-closure of M is defined to be the set of all points x in X each of whose closed neighborhoods meets M; M is called \(\theta\)-closed if M equals its \(\theta\)-closure. If P is a property of topologies, a P-space is called P-\(\theta\)-closed (or P-closed) if the space is a \(\theta\)-closed (closed) subset of every P-space in which it can be embedded as a subspace. The author study P-\(\theta\)-closed spaces for certain properties P between Hausdorff and regular, in particular, for \(P=S(n)\), where n denotes a positive integer. For \(n=0,1\), or 2, \(S(n)=T_ 1\), Hausdorff, or Urysohn, respectively, where by a Urysohn space one means a space in which any two distinct points have disjoint closed neighborhoods. More generally, for \(n>0\), a space X is called an S(n)- space provided that whenever x, y are distinct points of X there exists a sequence of open neighborhoods of x, \(U_ 1,U_ 2,...,U_ m\), such that each \(U_{i+1}\supset_{cl}U_ i\) and \(y\not\in_{cl}U_ n.\)
Some of the results obtained are the following. P-\(\theta\)-closedness implies P-closedness, and compactness plus P-\(\theta\)-closedness. Every Hausdorff-\(\theta\)-closed space is compact. The product of an H-closed S(n)-space and an S(n)-closed space is S(n)-closed, and the product of a compact space and an S(n)-\(\theta\)-closed space is S(n)-\(\theta\)-closed. Characterizations of S(n)-closed and S(n)-\(\theta\)-closed spaces are given, and a number of examples are given concerning relationships among these concepts and the extent to which they are hereditary, productive, or minimal. The authors conclude by posing a number of open questions.

Reviewer: R.M.Stephenson

MSC:

54D25	“\(P\)-minimal” and “\(P\)-closed” spaces
54D10	Lower separation axioms (\(T_0\)–\(T_3\), etc.)
54B10	Product spaces in general topology
54A10	Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)

Keywords:

\(\theta\)-closure; P-\(\theta\)-closed spaces; Urysohn space; Hausdorff- \(\theta\)-closed space; S(n)-\(\theta\)-closed space

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