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. For questions about Riemannian metrics use the tag (riemannian-geometry) instead.
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Is there 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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Is there a categorical definition of submetry?
(Updated to include effective epimorphism.)
This question is prompted by the recent discussion of why analysts don't use category theory. It demonstrates what happens when an analyst tries to use ...
user31373
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What should be the intuition when working with compactness?
I have a question that may be regarded by many as duplicate since there's a similar one at MathOverflow.
In $\mathbb{R}^n$ the compact sets are those that are closed and bounded, however the guy who ...
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$\pi$ in arbitrary metric spaces
Whoever finds a norm for which $\pi=42$ is crowned nerd of the day!
Can the principle of $\pi$ in euclidean space be generalized to 2-dimensional metric/normed spaces in a reasonable way?
For ...
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When is the closure of an open ball equal to the closed ball?
It is not necessarily true that the closure of an open ball $B_{r}(x)$ is equal
to the closed ball of the same radius $r$ centered at the same point $x$. For a quick example, take $X$ to be any set ...
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$A$ and $B$ disjoint, $A$ compact, and $B$ closed implies there is positive distance between both sets.
Claim: Let $X$ be a metric space. If $A,B\subset X$ are disjoint, $A$ is compact, and $B$ is closed, then there is $\delta>0$ so that $ |\alpha-\beta|\geq\delta\;\;\;\forall\alpha\in A,\beta\in B$.
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Continuous mapping on a compact metric space is uniformly continuous
I am struggling with this question:
Prove or give a counterexample: If $f : X \to Y$ is a continuous mapping from a compact metric space $X$, then $f$ is uniformly continuous on $X$.
Thanks for ...
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Connected metric spaces with disjoint open balls
Let $X$ be the $S^1$ or a connected subset thereof, endowed with the standard metric. Then every open set $U\subseteq X$ is a disjoint union of open arcs, hence a disjoint union of open balls.
Are ...
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Not every metric is induced from a norm
I have studied that every normed space $(V, \lVert\cdot \lVert)$ is a metric space with respect to distance function
$d(u,v) = \lVert u - v \rVert$, $u,v \in V$.
My question is whether every metric ...
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Every subsequence of $x_n$ has a further subsequence which converges to $x$. Then the sequence $x_n$ converges to $x$.
Is the following true?
Let $x_n$ be a sequence with the following property: Every subsequence of $x_n$ has a
further subsequence which converges to $x$. Then the sequence $x_n$ converges to $x$.
I ...
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Difference between metric and norm made concrete: The case of Euclid
This is a follow-up question on this one. The answers to my questions made things a lot clearer to me (Thank you for that!), yet there is some point that still bothers me.
This time I am making ...
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Difference between complete and closed set
What is the difference between a complete metric space and a closed set?
Can a set be closed but not complete?
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Continuous functions do not necessarily map closed sets to closed sets
I found this comment in my lecture notes, and it struck me because up until now I simply assumed that continuous functions map closed sets to closed sets.
What are some insightful examples of ...
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Why is it that $\mathbb{Q}$ cannot be homeomorphic to _any_ complete metric space?
Why is it that $\mathbb{Q}$ cannot be homeomorphic to any complete metric space?
Certainly $\mathbb{Q}$ is not a complete metric space. But completeness is not a topological invariant, so why is the ...
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Show that the countable product of metric spaces is metrizable
Given a countable collection of metric spaces $\{(X_n,\rho_n)\}_{n=1}^{\infty}$. Form the Cartesian Product of these sets $X=\displaystyle\prod_{n=1}^{\infty}X_n$, and define $\rho:X\times X\...
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