# Tag Info

### A complete metric space contains a convergent sequence or an infinite discrete subset

I. With Ramsey's theorem. Ramsey's theorem says that, for any positive integer $r$, if the $r$-element subsets of an infinite set are colored with two colors, then there is an infinite subset whose $r$...
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### A complete metric space contains a convergent sequence or an infinite discrete subset

The claim in the question can be shown using  the infinite version of Ramsey theorem. I.e., we are using the following fact: If $A$ is an infinite sets and all 2-element subsets of $A$ are colored ...
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### Can the twice-punctured plane be given a homogeneous metric?

Not a complete answer. Let me restrict attention to Riemannian metrics; I don't know what the non-Riemannian case looks like. If a Riemannian surface is homogeneous then in particular it must have ...
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Accepted

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1 vote

### Prove that there are no onto functions like $f:[0,1]\rightarrow[0,1]$ such that for each $x\in[0,1]$, $f^{-1}(x)$ is open.

Do we really need the assumption that $f$ is surjective to derive contradiction? It seems to me that any function $f\colon [0,1]\to[0,1]$ where $f^{-1}(x)$ is open for all $x\in [0,1]$ must be ...
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1 vote

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### Understanding the Proof of Relative Openness Theorem in Rudin's Principles of Mathematical Analysis

This may be an approach to consider (expanding somewhat on the comment by @MPW). Focusing on the "right-to-left" part of the proof: Suppose $G$ is open in $X.$ This means that every point ...
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1 vote

### Understanding the Proof of Relative Openness Theorem in Rudin's Principles of Mathematical Analysis

The subspace topology is usually defined as the statement given in the proposition. Rudin however defines "$E$ open relative to $Y$" in the context of metric spaces. The definition provided ...
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1 vote

### Understanding the Proof of Relative Openness Theorem in Rudin's Principles of Mathematical Analysis

First off, it's important to understand what's being proved here is precisely that the "subspace topology" is $\textit{equal}$ to Rudin's definition of being "relatively open" (e.g....
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1 vote

### Manhattan metric proof

The idea behind your proof is fine, but you can adjust some of the wording to avoid confusion. For instance, do not start with your equation (1); this is what you are trying to prove. Start by ...
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