Spectral reflections of topological spaces (Q1686704)

This paper deals with the relationship between the category \(\mathsf{Top}\) of topological spaces and continuous maps and its reflective subcategory \(\mathsf{Spec}\) of spectral spaces and spectral maps. Specifically, it addresses the following question(s): Let \(\mathbf{L} : \mathsf{Top} \to \mathsf{Spec}\) be the (spectral) reflector and \(\mathbf{S}_X : X \to \mathbf{L}(X)\) the reflection map of a topological space \(X\). How do properties of a topological space \(X\) (resp. of a continuous map \(f : X \to Y\)) relate to properties of its spectral reflection \(\mathbf{L}(X)\) (resp. \(\mathbf{L}(f) : \mathbf{L}(X) \to \mathbf{L}(Y)\))? After a few sections of preparatory nature, with basic terminology and notation concerning spectral spaces, basic facts about some reflective subcategories of \(\mathsf{Top}\), and basic properties of the spectral reflection map of a space, it is shown first how properties of \(\mathbf{L}(f)\), such as being surjective or an embedding or having a dense image, correspond to properties of \(f\). Then, it is proved that \(\mathbf{S}_X\) is a homeomorphism if and only if \(X\) is a Noetherian spectral space (from which it follows immediately that the sequence of iterated spectral reflections of a non-Noetherian space will never stabilize). Separation properties and notions related to compactness are also treated. In particular, it is proved that {\parindent=0.7cm\begin{itemize}\item[(1)] \(X\) is \(T_0\) iff \(\mathbf{S}_X\) is an embedding, \item[(2)] \(X\) is \(T_2\) (Hausdorff) iff for any \(x\neq x'\) in \(X\), \(\mathbf{S}_X(x)\) and \(\mathbf{S}_X(x')\) do not have a common generalization in \(\mathbf{L}(X)\),\item[(3)] \(X\) is \(T_4\) iff \(\mathbf{L}(X)\) is a \(T_4\)-space,\item[(4)] \(X\) is compact Hausdorff iff \(\mathbf{L}(X)\) is \(T_4\) and \(\mathbf{S}_X(X)\) is the space of closed points of \(\mathbf{L}(X)\). \end{itemize}} Finally, it is shown that the spectral reflection provides a natural construction of the Stone-Čech compactification of a completely regular space and of the Gleason cover of a compact Hausdorff space.