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Let $f : X \to Y$ and $g : Y \to Z$ be continuous maps (between topological spaces). Assume these hypotheses: $f : X \to Y$ is a split surjection, i.e. has a section. $g \circ f : X \to Z$ is a loc …

Allow me to use some non-standard terminology: A h-contractible space is a non-empty topological space $X$ such that, for any topological space $T$ and any pair of continuous maps $f_0, f_1 : T \to …

Let $f : X \to Y$ be an open (and continuous) map of locales. Suppose the relative diagonal $\Delta_f : X \to X \times_Y X$ is an open embedding of locales. Does it follow that $f : X \to Y$ is a loca …

No. In fact any full subcategory of $\mathbf{Top}$ that contains all the discrete spaces cannot be monadic over $\mathbf{Set}$ unless it contains only discrete spaces. Indeed, for any such subcategory …

Let $X$ be a topological space. Consider the functor $P^X : \textbf{Set} \to \textbf{Set}$ that sends each set $Y$ to the set of continuous maps $X \to Y$. It is not hard to check that $P^X : \textbf{ …

Let $\textbf{Top}$ be the category of all topological spaces, including the bad ones. We can make $\textbf{Top}$ into a simplicially enriched category as follows: Given topological spaces $X$ and $Y$ …

It is well known that: Theorem. For a locale (resp. topological space) $X$, the following are equivalent: $X$ is compact, i.e. every open cover of $X$ has a finite subcover. For every locale (resp. …

It turns out that Vermeulen has essentially answered the question in [A note on stably closed maps of locales]. The argument there implies: Theorem. Let $g : X \to S$ be a morphism of locales. The fol …