# 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.

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### Does there exist 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 ...
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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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### 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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### 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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### 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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### 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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### 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 $A,B\in X$ are disjoint, if A is compact, and if B is closed, then $\exists \delta>0: |\alpha-\beta|\geq\delta\;\;\;\forall\alpha\in A,\beta\in B$. Proof. ...
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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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### 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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### 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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### If every real-valued continuous function is bounded on $X$ (metric space), then $X$ is compact.

Let $X$ be a metric space. Prove that if every continuous function $f: X \rightarrow \mathbb{R}$ is bounded, then $X$ is compact. This has been asked before, but all the answers I have seen prove the ...
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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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### Nonobvious examples of metric spaces that do not work like $\mathbb{R}^n$

This week, I come to the end of the first year analysis, and suffer from a "crisis of motivation." With this question, I want to chase away my thought, "Why is it important to study the general ...
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### Homeomorphic to the disk implies existence of fixed point common to all isometries?

A fellow grad student was working on this seemingly simple problem which appears to have us both stuck. (The problem naturally came up in his work so isn't from the literature as far as we know). Let ...
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### The category of compact metric spaces

Let us denote by $(\mathrm{CompMet})$ the category of compact metric spaces with Lipschitz maps as morphisms. I'm interested in properties of this category. It seems to me that it has finite products (...
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### If $d(x,y)$ is a metric, then $\frac{d(x,y)}{1 + d(x,y)}$ is also a metric

Let $(X,d)$ be a metric space and for $x,y \in X$ define $$d_b(x,y) = \dfrac{d(x,y)}{1 + d(x,y)}$$ a) show that $d_b$ is a metric on $X$ Hint: consider the derivative of $f(t)$ = $\dfrac{t}{1+t}$ ...
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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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### Does there exist a space filling curve which sends every convex set to a convex set ?

Does there exist a surjective continuous function $f:[0,1]\to [0,1]^2$ which maps every convex set to a convex set ?
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### Existence of a Riemannian metric inducing a given distance.

Let $M$ be a smooth, finite-dimensional manifold. Suppose $M$ is also a metric space, with a given distance function $d: M \times M \rightarrow \mathbb{R}_{+}$, which is compatible with the original (...
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### Connected metric spaces with at least 2 points are uncountable.

That's a problem I proved (quite a while back) in tiny Rudin. However, I don't really get it. The other questions were actually useful results - I don't think I've ever come near using this result. ...
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### Show that in a discrete metric space, every subset is both open and closed.

I need to prove that in a discrete metric space, every subset is both open and closed. Now, I find it difficult to imagine what this space looks like. I think it consists of all sequences containing ...
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### Why do we want complete spaces? We don't we just use closed spaces?

Why do we care about the notion of a space being complete? Why don't just consider closed spaces? If the space is closed we know that the limits of a sequence exist and are in the set which is a ...
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### Difference between closed, bounded and compact sets

In real analysis, there is a theorem that a bounded sequence has a convergent subsequence. Also, the limit lies in the same set as the elements of the sequence, if the set is closed. Then when ...
My question is related with the definition of Cauchy sequence As we know that a sequence $(x_n)$ of real numbers is called Cauchy, if for every positive real number ε, there is a positive integer $N \... 4answers 20k views ### Union of closure of sets is the closure of the union: true for finite, false for infinite unions Let$A_i$be a subset of a metric space for each$i\in \mathbb{N}$. Let$B_n := \bigcup_{i=1}^n A_i$. Prove (for any)$n \in \mathbb{N}$that$\overline{B_n} = \bigcup_{i=1}^n \overline{A_i}$. ... 4answers 2k views ### Proving that the triangle inequality holds for a metric on$\mathbb{C}$Show that$(X,d)$is a metric space where$X =\Bbb C $and the distance function is defined as: $$d(x,y) = \frac {2|x-y|}{\sqrt {1+|x|^2} + \sqrt {1 + |y|^2}}, \text{ for } x,y \in \Bbb C.$$ I ... 3answers 4k views ### Continuity of the function$x\mapsto d(x,A)$on a metric space Let$(X,d)$be a metric space. How to prove that for any closed$A$a function$d(x,A)$is continuous - I know that it is even Lipschitz continuous, but I have a problem with the proof:$$|d(x,a) - ... 2answers 12k views ### Prove: Every compact metric space is separable How to prove that Every compact metric space is separable$?$Thanks in advance!! 4answers 21k views ### Cauchy sequence is convergent iff it has a convergent subsequence Prove that if$\left ( x_{n} \right )$is a Cauchy sequence in a metric space X then$\left ( x_{n} \right )$is convergent if and only if$\left ( x_{n} \right )$has a convergent subsequence. Note: ... 2answers 7k views ### Understanding the idea of a Limit Point (Topology) I have attached an image of how I was visualizing a limit point, but I'm now not so sure that I have understood the concept correctly after attempting to really draw out what I was visualizing. I'll ... 2answers 719 views ### Can you define arc length using a piece of string? In calculus, how we calculate the arc length of a curve is by approximating the curve with a series of line segments, and then we take the limit as the number of line segments goes to infinity. This ... 1answer 10k views ### Difference between an isometric operator and a unitary operator on a Hilbert space It seems that both isometric and unitary operators on a Hilbert space have the following property:$U^*U = I$($U$is an operator and$I$is an identity operator,$^*$is a binary operation.) What ... 3answers 7k views ### Real numbers equipped with the metric$ d (x,y) = | \arctan(x) - \arctan(y)| $is an incomplete metric space I have to show that the real numbers equipped with the metric$ d (x,y) = | \arctan(x) - \arctan(y)| $is an incomplete metric space. Certainly, I have to search for a Cauchy sequence of real numbers ... 4answers 2k views ### How to develop an intuitive feel for spaces I'm a physicist who's currently delving deeper into what I would call more 'hardcore' maths (e.g. FEM and control theory). Every now and then, I come across various spaces, such as vector spaces, ... 1answer 531 views ### Is$\mathbb{Q}^2$homeomorphic to$\mathbb{Q}^2\setminus \{0\}$? I know that$\mathbb{R}^2$and$\mathbb{R}^2\setminus\{(0,0)\}$are not homeomorphic. (For examle$\pi_1(\mathbb{R}^2)=\{e\}$, but$\pi_1(\mathbb{R}^2\setminus\{(0,0)\})=\mathbb{Z}$). But what can ... 5answers 17k views ### Showing that if a subset of a complete metric space is closed, it is also complete Let$(X, d(x,y))$be a complete metric space. Prove that if$A\subseteq X$is a closed set, then$A$is also complete. My attempt: I tried to prove that every Cauchy sequence$(b_n)$of points of$A$... 4answers 2k views ### Developing the unit circle in geometries with different metrics: beyond taxi cabs My class had a good time redeveloping the unit circle under the taxicab metric. Now some of them want to do it again with another similar metric. I want to give this to some of my "honors" high-school*... 1answer 527 views ### Is every connected subset of the Sierpiński triangle arcwise connected? I think this should be true. If it's indeed the case, it seems like this should be a known result, so references are welcome. I managed to prove that (assuming$S$is the connected subset)$S$minus ... 2answers 1k views ### What is a metric for$\mathbb Q$in the lower limit topology? A useful source of counterexamples in topology is$\mathbb R_\ell$, the set$\mathbb R$together with the lower limit topology generated by half-open intervals of the form$[a,b)$. For example this ... 5answers 11k views ### What is the difference between metric spaces and vector spaces? Does a metric space have an origin? That is, does it have$(0,0)$. Does a vector space have an origin? It seems whatever you can do in a metric space can also be done in a vector space. Is this true?... 4answers 9k views ### Equivalent metrics determine the same topology Suppose that there are given two distance functions$d(x,y)$and$d_1 (x,y)$on the same space$S$. They are said to be equivalent if they determine the same open sets. Show that$d$and$d_1$are ... 6answers 2k views ### Why do we use the Euclidean metric on$\mathbb{R}^2$? On the train home, I thought I would try to prove$\pi$is irrational. I needed a definition, so I used:$\pi$is the area of the unit circle. But what is a circle? A circle is the set of tuples$(...
From Wikipedia: If $M$ is a metric space with metric $d$, and $\sim$ is an equivalence relation on $M$, then we can endow the quotient set $M/{\sim}$ with the following (pseudo)metric. Given ...