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