A direct approach to \(K\)-reflections of \(T_0\) spaces. (English) Zbl 1434.54007
The author presents a direct approach to \(\mathbf{K}\)-reflections of \(T_0\) spaces.
For a full subcategory \(\mathbf{K}\) of \(\mathbf{Top}_0,\) containing \(\mathbf{Sob}\) (that is, the category of sober spaces), and a \(T_0\) space \(X,\) let \(\mathbf{K}(X)=\{A\subseteq X:A\) is closed and for any continuous mapping \(f:X\rightarrow Y\) to a \(\mathbf{K}\)-space \(Y,\) there exists a unique \(y_A\in Y\) such that \(\overline{f(A)}=\overline{\{y_A\}}\}\). Equip \(\mathbf{K}(X)\) with the lower Vietoris topology and denote the resulting space by \(P_H(\mathbf{K}(X))\).
The author proves that if \(P_H(\mathbf{K}(X))\) is a \(\mathbf{K}\)-space, then the pair \((X^k=P_H(\mathbf{K}(X)),\eta_X)\), where \(\eta_X:X\rightarrow X^k\), \(x\mapsto \overline{\{x\}},\) is the \(\mathbf{K}\)-reflection of \(X\).
Now \(\mathbf{K}\) is called an adequate category if for any \(T_0\) space \(X,\) \(P_H(\mathbf{K}(X))\) is a \(\mathbf{K}\)-space. It follows that if \(\mathbf{K}\) is adequate, then \(\mathbf{K}\) is reflective in \(\mathbf{Top}_0\).
It is shown that the three well-known categories \(\mathbf{Sob}\), \(\mathbf{Top}_d\) of \(d\)-spaces and \(\mathbf{Top}_w\) of well-filtered spaces are all adequate. The same applies to what the author calls a Keimel-Lawson category.
Some important properties of \(\mathbf{K}\)-spaces and \(\mathbf{K}\)-reflections of \(T_0\) spaces are investigated. Among other things, it is shown that if \(\mathbf{K}\) is adequate, then for each finite family \(\{X_i:1\leq i\leq n\}\) of \(T_0\) spaces, we have that \((\Pi_{i=1}^nX_i)^k=\Pi_{i=1}^n{X^k_i}\) (up to homeomorphism).
For a full subcategory \(\mathbf{K}\) of \(\mathbf{Top}_0,\) containing \(\mathbf{Sob}\) (that is, the category of sober spaces), and a \(T_0\) space \(X,\) let \(\mathbf{K}(X)=\{A\subseteq X:A\) is closed and for any continuous mapping \(f:X\rightarrow Y\) to a \(\mathbf{K}\)-space \(Y,\) there exists a unique \(y_A\in Y\) such that \(\overline{f(A)}=\overline{\{y_A\}}\}\). Equip \(\mathbf{K}(X)\) with the lower Vietoris topology and denote the resulting space by \(P_H(\mathbf{K}(X))\).
The author proves that if \(P_H(\mathbf{K}(X))\) is a \(\mathbf{K}\)-space, then the pair \((X^k=P_H(\mathbf{K}(X)),\eta_X)\), where \(\eta_X:X\rightarrow X^k\), \(x\mapsto \overline{\{x\}},\) is the \(\mathbf{K}\)-reflection of \(X\).
Now \(\mathbf{K}\) is called an adequate category if for any \(T_0\) space \(X,\) \(P_H(\mathbf{K}(X))\) is a \(\mathbf{K}\)-space. It follows that if \(\mathbf{K}\) is adequate, then \(\mathbf{K}\) is reflective in \(\mathbf{Top}_0\).
It is shown that the three well-known categories \(\mathbf{Sob}\), \(\mathbf{Top}_d\) of \(d\)-spaces and \(\mathbf{Top}_w\) of well-filtered spaces are all adequate. The same applies to what the author calls a Keimel-Lawson category.
Some important properties of \(\mathbf{K}\)-spaces and \(\mathbf{K}\)-reflections of \(T_0\) spaces are investigated. Among other things, it is shown that if \(\mathbf{K}\) is adequate, then for each finite family \(\{X_i:1\leq i\leq n\}\) of \(T_0\) spaces, we have that \((\Pi_{i=1}^nX_i)^k=\Pi_{i=1}^n{X^k_i}\) (up to homeomorphism).
Reviewer: Hans Peter Künzi (Rondebosch)
MSC:
| 54D35 | Extensions of spaces (compactifications, supercompactifications, completions, etc.) |
| 06B35 | Continuous lattices and posets, applications |
| 06F30 | Ordered topological structures |
| 18F60 | Categories of topological spaces and continuous mappings |
Keywords:
\(K\)-reflection; \(K\)-set; sober space; well-filtered space; \(d\)-space; Keimel-Lawson categoryReferences:
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