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Compactness defined by an epireflector. (English) Zbl 0773.54014

Topology, Proc. 5th Int. Meet., Lecce/Italy 1990, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 29, 575-585 (1992).
[For the entire collection see Zbl 0754.00013.]
The author studies Stramaccia’s \(r\)-closure and \(r\)-compactness [L. Stramaccia, Topology, 3rd Natl. Meet., Trieste/Italy 1986, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 18, 423-432 (1988; Zbl 0655.54009)] with respect to epireflectors of the form \(\text{Top}\overset {r} \rightarrow R\). The \(r\)-closure of \(M\subseteq X\) is \(r^{-1}_ X\overline{[r_ X[M]]}\) (where \((\overline{\phantom{rX}})\) denotes usual closure). \(X\) is called \(r\)-compact provided that the topology (denoted by \(X_ r\)) that is induced by the \(r\)-closure (and is called the \(r\)-topology) is compact. If \(Y\subseteq X\) it is shown that the \(r\)-topology on \(Y\) is finer than (and can be strictly finer than) the subspace topology on \(Y\) induced by \(X_ r\). It is shown that a space \(X\) is \(r\)-compact if for each space \(Y\) the second projection \(X\times Y\to Y\) preserves \(r\)- closed sets and that the converse is true if \(r\) preserves products of pairs. If \(q(r)\) denotes the reflector from Top to the quotient reflective hull of \(R\) then it is shown that \(X\) is \(q(r)\)-compact iff it is \(r\)-compact and \(rX\cong q(r)X\).

MSC:

54D30 Compactness
54B30 Categorical methods in general topology
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