# Tagged Questions

Metric spaces are sets on which a metric is defined. A metric is a generalisation of the concept of "distance". Metric spaces should not be confused with topological spaces.

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### $M,N$ metric spaces, $\phi:M\to N$ a surjective open map. Show that the map $f:N\to P$ is continuous iff $f\circ \phi$ is continuous

I need to show the following: $M,N$ metric spaces, $\phi:M\to N$ a surjective open map. Show that the map $f:N\to P$ is continuous iff $f\circ \phi$ is continuous In order to show that the composite ...
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### boundary of $\{x\in M: f(x)>0\}\implies f(x)=0$

Given a continuous function $f:M\to \mathbb{R}$, and $A=\{x\in M: f(x)>0\}$, I need to show that if $x\in \partial A$(boundary of $A$), then $f(x) = 0$. I know that $\partial A$ is the set of all ...
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### Relation between Compactness, Closedness and Completness of metric spaces

I would like to know as many relations as possible to get a better picture. I know that if $f$ is continuous and $(X,d)$ is complete, then $f(X)$ is complete $\iff$ closed. Question:However, are ...
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### Separable metric spaces that are not normable

Not quite sure whether this question belongs here or on MESE. Anyway: Can anyone suggest a good example of a separable metric space that is neither normable nor a subset of normed space with the ...
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### $\mathbb{R}^2$ to $\mathbb{R}^1$ Injective Mapping While Preserving the Triangle Inequality

Is there a way to map from $\mathbb{R}^2$ to $\mathbb{R}^1$, such that every point in $\mathbb{R}^2$ has a unique point in $\mathbb{R}^1$ and you preserve the distance (isometry) relations of ...
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### Topological spaces without homeomorphisms?

Can we find a topological space which is not homeomorphic to any other? Of course, not considering the space itself neither the empty set. And if's so, is it possible to classify them? Just like the ...
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### Proper map and sequences in metric spaces

Let $f:X\to Y$ be a continuous map between metric spaces satisfying the Heine-Borel theorem. Show that $f$ is proper if the following condition holds: For every sequence $x_n\in X$ such that ...