Every scattered space is subcompact (Q390392)

In this paper all spaces are assumed to be Tychonoff. If \(X\) is a space then \(\tau (X)\) is its topology and \(\tau^{\ast}(X)=\tau (X)\setminus \{\emptyset\}\).NEWLINENEWLINELet \(Y\) be a space. A family \(\mathcal{U}\subset \tau^{\ast}(Y)\) is called a regular filter base if, for any \(U,V\in \mathcal{U}\) there is \(W\in \mathcal{U}\) such that \(Cl(W)\subset U\cap V\). The space \(Y\) is subcompact if it has a base \(\mathcal{B}\subset \tau^{\ast}(Y)\) such that every regular filter base \(\mathcal{U}\subset\mathcal{B}\) has non-empty intersection.NEWLINENEWLINEA space \(X\) is called scattered if every non-empty subspace of \(X\) has an isolated point. Also, a space \(X\) is called Čech-complete if it is homeomorphic to a dense \(G_{\delta}\)-subset of a compact space.NEWLINENEWLINEIn this paper the authors prove that:NEWLINENEWLINE(1) Every scattered space is hereditarily subcompact.NEWLINENEWLINE(2) Every scattered metrizable space is Čech-completeNEWLINENEWLINE(3) For any countable space \(X\), the following properties are equivalent:NEWLINENEWLINE(a) \(X\) is hereditarily subcompact;NEWLINENEWLINE(b) \(X\) is subcompact;NEWLINENEWLINE(c) \(X\) is scattered.NEWLINENEWLINE(4) Any finite union of subcompact spaces is subcompact.NEWLINENEWLINE(5) If \(X\) is a linearly ordered compact space, \(A\) is a countable subset of \(X\) then \(Y=X\setminus A \) is subcompact.NEWLINENEWLINE(6) Suppose that \(D\not = \emptyset\) is a discrete space and \(I\) is a non-empty set. Then any dense \(G_{\delta}\)-subspace of \(D^I\) is subcompact.NEWLINENEWLINEFinally, the authors give some interesting problems in this area.