Let A be a finite metric space .I want to prove that every subset of A is open. I let the set B, be any subset of A. Since A is finite,then I know that A/B is also finite.I'm stuck here how can this help me reach to a proof? I beg your help
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Every finite metric space is equivalent to a discrete space. |
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A space is discrete iff every singleton set is open. If M is a finite metric space and $x\in M$. Let $\epsilon$ be the minimum distance from x to other points of M, the $B_{\epsilon}(x)$ contains x only So $\{x\}$ is open for every x.So M is discrete. |
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Massive hint: In a metric space, finite point sets are closed. So suppose that you have a subset $B$ of $A$. Then $A \setminus B$ is a finite point set so..... |
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Hint: If $(A,d)$ is a finite metric space and $x \in A$ and we let $$\delta=\min_{y \in A \setminus \{x\}}d(x,y)$$ then what is in $B(x,\delta)$? |
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