Metric spaces are sets on which you can measure the "distance" between any two points. The distance measurement is generally required to be symmetric (so distance from $A$ to $B$ is the same as distance from $B$ to $A$), positive for two distinct points, and obeying the triangle inequality.

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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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How many metrics are there on a set up to topological equivalence?

I want to find the number of topologically nonequivalent metrics on a set. I think if the cardinal of set is finite then we have one metric that is the discrete metric and every metric on this set ...