# Tagged Questions

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### Category of metric spaces versus category of non-empty spaces

Denote by $\mathbf{Met}$ the category of metric spaces with metric maps as morphisms. A function $(X,d)\xrightarrow{\ f\ }(X',d')$ is called metric if for every pair of points $x,y\in X$ we have ...
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### Can I take Inverse Limits as Cauchy sequences literally?

I have been told to think of inverse limits as "Cauchy Completions" under some metric, for instance through the construction of the p-adic numbers. This got me thinking, though, and I wonder if the ...
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### Completeness in a Category of Metric Spaces.

Is there a way to describe completeness within a category of metric spaces? The point is that I'd like to have a description of compactness in metric spaces by something of the form totally bounded + ...
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### What's the real name for these things? Categories whose morphisms have “length.”

A fairly obvious "categorification" of metric spaces is as follows. First, let us agree to view $\mathbb{R}_+$ as an ordered Abelian monoid, where by "Abelian monoid" we really mean a category whose ...
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### Free Metric Space?

Do free metric spaces exist? Ie.: An object in the category of metric spaces and lipschitzian maps. If so would these be the complete metric spaces, since they satisfy a similar universal property? ...
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### Is there a category such that if $\mathbb{R}$ is viewed as an object, we have that $\mathbb{R}^2$ is equipped with the Euclidean distance function?

Viewing $\mathbb{R}$ as an object of the category of metric spaces and metric maps, we have that $\mathbb{R}^2$ is equipped with the distance function d(x,y) = ...
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### Does there exist another way of obtaining a topological space from a metric space equally deserving of the term “canonical”?

Every metric space is associated with a topological space in a canonical way. According to this source, this amounts to a full functor from the category of metric spaces with continuous maps to the ...
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### pseudo-inverse to the operation of turning a metric space into a topological space

The construction of turning a metric space $(X,d)$ into a topological space by inducing the topology generated by the open balls gives rise to a functor $Met\to Top$ for any reasonable category $Met$ ...
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### Is there a categorical definition of submetry?

(Updated to include effective epimorphism.) This question is prompted by the recent discussion of why analysts don't use category theory. It demonstrates what happens when an analyst tries to use ...
I need to prove that category $\mathrm{Met}$ of metric spaces and continuous maps doesnt possess uncountable product of non-one point spaces. Definition. A pair $(X,\{\pi_\nu:\nu\in\Lambda\})$ where ...
Let us denote by $(\mathrm{CompMet})$ the category of compact metric spaces with Lipschitz maps as morphisms. I'm interested in properties of this category. It seems to me that it has finite products ...