# 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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### $X$ is metric space s.t. for every metric space $Y$ and any continuous function $f : X \to Y$ , $f(X)$ is closed in $Y$ ; is $X$ compact?

Let $X$ be a metric space such that for every metric space $Y$ and any continuous function $f : X \to Y$ , $f(X)$ is closed in $Y$ , then is $X$ compact ? Compare with this $A \subseteq \mathbb R^n$...
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### Proof about a Topological space being arc connect

While reading a book i found a topological space described as: Let $(X,\tau)$ be the topological space formed by adding to the ordinary closed unit interval $[0,1]$ another right end point,say ...
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### Continuous image of a cantor set and other space filling curves

I am studying the theory of space filling curves, more specifically looking at the continuous image of a Cantor set. I have been given this definition to characterise the continuous image of a Cantor ...
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### Limit point - contains one, or infinitely many points of set?

I'm reading through Functional Analysis by Bachman. He defines a limit point as follows: The point $x$ is said to be a limit point of $A \subset X$ iff for every $r$, $S_r(x) \cap A$ contains ...
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### Derivation of metric from product topology?

Suppose you have two topological spaces, g11 and g22, which are "components" of a more general topology. For example, suppose that a metric has components g11, g12, g21, and g22. And suppose you want ...
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### $f(x)=0$ for $x\in X$, then $f(x) = 0$ for $x\in\overline{X}$

I'm trying to prove: $f(x)=0$ for $x\in X$, then $f(x) = 0$ for $x\in\overline{X}$ where $f$ is a function from $\mathbb{R}$ to $\mathbb{R}$ and is continuous. I know that the set $\overline{X}$ is ...
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### Weak measurability of a set-valued map

Suppose that $A$ and $B$ are compact metric spaces. Let $f:A\times B\to B$ be a Borel measurable map (in the sense that for every Borel set $S\subseteq B$, $f^{-1}(S)$ belongs to the $\sigma$-algebra ...
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### Base of the Baire space [closed]

Why base of the Baire space is countable? Because it is a perfect Polish space, which means it is a completely metrizable second countable space with no isolated points.
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### Computing Hausdorff metric for some sets

Just started to learn about metric spaces, and I came across the Hausdorff metric. Let $K$ be the family of non-empty closed subsets of $[0,1]$. For $A \in K$ and $\delta > 0$ let $A_{\delta}$ be ...
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### $S := \{x \in \Bbb R^3: ||x||_2 = 1 \}$ and $T: S^2 \to \Bbb R$ is a continuous function. Is $T$ injective?

$S := \{x \in \Bbb R^3: ||x||_2 = 1 \} \subset (\Bbb R^3, || \cdot ||_2 )$ and $T:S^2 \to (\Bbb R, |\cdot |)$ is a continuous function. I've already shown that T_{\mathrm{max}} := \mathrm{sup}\{ T(...
I was reading a proof that if a sequence of functions from $M$ to $N$, where $N$ is complete, converges uniformly in $X$, then they converge uniformly in $\overline{X}$, and it uses this result: $\... 1answer 69 views ### if$M$is compact, then every continuous bijection$F:M\to N$is an homeomorphism My book proves that: if$M$is compact, then every continuous bijection$f:M\to N$is an homeomorphism by the following: Being$f$closed, your inverse$g:N\to M$is a function such that$F\subset ...
Tychonoff's theorem: The cartesian product $M = \prod_{i=1}^{\infty}M_i$ is compact $\iff$ each $M_i$ is compact. My book, before proving it, says that the proof will happen like this: Given an ...