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Don't pick a convergent subsequence: pick a point in $ \cap_{n \in \mathbb{N}} \overline{\{ x_n,x_{n+1}, \dots \}}$ (the overline meaning closure) which is non empty as an intersection of decreasing f …

Note : In order to define the alexandrov compactification of $X$ you have to take the algebra of function converging at infinity (not just those converging to $0$) If by a compactification of $X$ you …

When $X$ is compact, it is discrete (hence not compact unless it is finite). For any function $f:X \to \{0,1\}$, then both $K_0 = f^{-1}(0)$ and $K_1=f^{-1}(1)$ are compact subset of $X$, and so the s …

Let $X$ be a locale or a topological space. $I$ denote the unit interval of the real numbers, and $X^I$ the space of functions from $I$ to $X$ (The locale exponential if $X$ is a locale or the set of …

By groupoid I mean "open topological groupoid",i.e. topological groupoids whose source and target maps are open surjections, and the notion of equivalence I'm considering is the isomorphism in the ca …

Is there a nice (topological) characterization of the locales such that the axiom of dependant choices holds in the internal logic of the topos of sheaves ? I would also be interested in the case of t …

I'm assuming that by "compact" you only mean which satisfies the finite cover properties, and not compact and Hausdorff. in this case the following produces a counterexample: Let $X_1$ be $\mathbb{R} …

Let me expand a bit my comment as this is a rather subtle property. As I said any locally compact Hausdorff topological space is a strongly hausdroff locally compact locales. (and under the axiom of …

The answer is yes. It appears for example as lemma C3.1.15 in Johnstone's sketches of an elephant. Roughly, it can be proved by working in the internal logic of the target (hence assuming that the t …

I have thought about this recently, and here is I think the best constructively valid statement one can extract from Brouwer fixe point theorem (framework : internal logic of an elementary topos, real …

I'm looking for a reference for the following result: Let $X$ be a locally compact Hausdorff topological space. let $\mathcal{R}$ be the sheaf of continuous functions with values in $\mathbb{R}$ over …

I definitely expect that there is much more than one good answer. But, here is one that one can get easily by just patching together several classical facts: The category of compact Hausdorf topolog …

For short, the exponential $(X,Y)$, characterized by the usual universal properties: morphisms from any locale $Z$ to $(X,Y)$ are functions from $X \times Z$ to $Y$, exists for all $Y$ if and only if …

If $X$ is locally compactly generated then $X$ is compactly generated because every locally compact space is compactly generated. So given $f:X \to Y$ a map such that $f\circ i$ is continuous for ever …

If the absence of adjoints is what worries you, you can consider this to be a two-step process - and I would argue that in practice this is the case in the vast majority of cases: One first generalize …