Relations of topologies as tools of (bi)topological applications (Q897453)

For a topological space \((X,\tau_1)\), \(A\subset X\) and \(A\not\in\tau_1\), the topology \(\tau_1(A)= \{U\cup(V\cap A): U,V\in\tau_1\}\) is said to be a simple extension of the topology \(\tau_1\); the bitopological space \((X,\tau_1< \tau_1(A))\) is of interest in the paper. For two topologies \(\tau_1\) and \(\tau_2\) on a set \(X\), \(\tau_1\text{\,imp\,}\tau_2\) iff (i) \(\tau_1\subset\tau_2\), \(\tau_1\neq\tau_2\); (ii) if \(\tau_1\subset\tau\subset\tau_2\) (for some topology \(\tau\) on \(X\)), then \(\tau_1=\tau\) or \(\tau_2=\tau\) and further, \(\tau_1\) and \(\tau_2\) are called adjacent iff \(\tau_1\text{\,imp\,}\tau_2\) or \(\tau_2\text{\,imp\,}\tau_1\). Primarily, the paper is interested in studying some properties of simple extensions and adjacent topologies; the interesting problem of passage of some important properties from \(\tau_1\) to \(\tau_1(A)\) is explored. Among the properties, as mentioned, the Blumberg property, pseudo-completeness, connectedness, submaximality, to be a Baire space, to be an \(H\)-closed space, to be an \(HD\)-space, to be an \(eD\)-space and to be a nodec space are worth mentioning. The concepts of \(HD\)-space and \(eD\)-space are introduced by the author; if, for each neighbourhood assignment \(\phi:X\to\tau\), there exists a closed discrete subset \(F\) of \((X,\tau)\) such that \(\bigcup \{\tau\text{\,cl\,} \phi(x): x\in F\}= X\), then \((X,\tau)\) is called an \(HD\)-space; if further, for each open neighborhood assignment \(\phi\), there are an open neighborhood assignment \(\psi:X\to\tau\) and a closed discrete subset \(F\) of \((X,\tau)\) such that \(\phi(x)\subset\psi(x)\), for each \(x\in X\) and \(\bigcup\{\psi(x): x\in F\}= X\), then \((X,\tau)\) is called an \(eD\)-space. It is proved that every countably compact \(HD\)-space is \(H\)-closed and \((X,\tau')\) is an \(eD\)-space for each topology \(\tau'\supseteq\tau\) on \(X\) when \((X,\tau)\) is a \(D\)-space. For a connected \(T_1\)-space \((X,\tau_1)\), \(A\neq\tau_1\) and \(\tau_1\text{\,imp\,}\tau_1(A)\), it is shown that \((X,\tau_1(A))\) is connected iff \(A\) is singular at a point \(a\) in \((X,\tau_1)\), where \(\{a\}= A\setminus\tau_1\text{\,int}(A)\). With the notations \(P_\tau(p)\) and \(F_\tau(p)\), where \(P_\tau(p)\) denotes the family of sets, singular at the point \(p\) and \(F_\tau(p)\) denotes an open filter \(\{U: p\in U\}\), it is proved that a necessary and sufficient condition for a \(T_1\)-space \((X,\tau_1)\) to be maximally connected is that the bitopological space \((X,\tau_1<\tau_2)\) is \((2,1)\) submaximal for any topology \(\tau_2\supset\tau_1\) on \(X\) and \(F_{\tau_1}(p)= P_{\tau_1}(p)\), for each \(p\in X\). A topological space \((X,\tau)\) is termed nodec if all nowhere dense subsets of \(X\) are closed. With \(H(X,\tau)\), as the class of all homeomorphisms of \((X,\tau)\) onto itself, it is shown that, for a nodec space \((X,\tau)\) without isolated points, the family \(\gamma=\{\emptyset\}\cup\{U\in\tau: X\setminus U\) is discrete\} is a topology on \(X\) and \(H(X,\tau)= H(X,\gamma)\) when every point of \(X\) has a neighborhood in \((X,\tau)\) which is not dense in \((X,\gamma)\).