# Questions tagged [metric-spaces]

Metric spaces are sets on which a metric is defined. A metric is a generalization of the concept of "distance" in the Euclidean sense. Metric spaces arise as a special case of the more general notion of a topological space.

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### Does there exist a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
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### a subclass of quasi metric spaces with properties almost identical to metric spaces

It is well known that passing from metric spaces to quasi-metric spaces (i.e., dropping the requirement that the metric function $d:X\times X\to \mathbb R$ satisfies $d(x,y)=d(y,x)$) carries with it ...
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### Contracting subsets

Let $X$ be a (locally finite) metric graph (all of whose edges are length 1). A subset $A \subset X$ is contracting if there exists a constant $C \geq 0$ such that the nearest point projection on $A$ ...
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### A generalization of the abc-conjecture?

For a natural number $a$ define $X_a := \{a/k| 1\le k \le a \}$. Then it is not difficult to show that $|X_a \cap X_b| = \gcd(a,b)$. Using this one can define similarities over the natural numbers, ...
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### A definition of differentiable functions for arbitrary topological spaces

Background It is well-known that there is no notion of derivative for arbitrary topological spaces. However while investigating the notion of derivative as we find in one variable real analysis I came ...
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### Hyperbolic metric spaces 2

I am trying to prove a lemma in Burago's "A Course in Metric Spaces" (Exercise 8.4.4, p.286). Here is a link to a different person's question about the very next exercise in that book, which also ...
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### Embedding whole flat 2-Torus in $\Bbb R^3$
I am trying to better understand an isometric embedding of a flat 2-Torus in $\Bbb R^3$ via a $C^1$ map. Here is a visualization involving $C^1$ fractals: This embedding is warped by an infinite ...