Metric spaces are sets on which you can measure the "distance" between any two points. The distance measurement is generally required to be symmetric (so distance from $A$ to $B$ is the same as distance from $B$ to $A$), positive for two distinct points, and obeying the triangle inequality.

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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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$\pi$ 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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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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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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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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Can a collection of points be recovered from its multiset of distances?

Consider $n$ distinct points $x_1,\dots,x_n$ on $\mathbb{R}$. Associated to these points is the multiset of all distances $d(x_i,x_j)$ between two points. Suppose one is only handed this multiset (you ...
26
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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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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 ...
20
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A strangely connected subset of $\Bbb R^2$

Let $S\subset{\Bbb R}^2$ (or any metric space, but we'll stick with $\Bbb R^2$) and let $x\in S$. Suppose that all sufficiently small circles centered at $x$ intersect $S$ at exactly $n$ points; if ...
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When is a metric space Euclidean, without referring to $\mathbb R^n$?

Normally, the Euclidean space is introduced as $\mathbb R^n$. However, I've now been thinking about how one might define the $n$-dimensional Euklidean space only from the properties of the metric. ...
19
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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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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 ...
19
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Homeomorphism preserving distance

I have a problem but I don't know if there is a solution or a counter-example. Problem: Let $M$ be a non trivial compact connected metric space and let $f:M\to M$ be a homeomorphism. Show that there ...
19
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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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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?
17
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650 views

If every open set is a countable union of balls, is the space separable?

Suppose we have a metric space in which every open set is expressible as a countable union of balls. Is this space necessarily separable? Thank you.
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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) - ...
16
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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 ...
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Two metrics induce the same topology, but one is complete and the other isn't

I'm looking for an example of two metrics that induce the same topology, but so that one metric is complete and the other is not (Since it is known that completeness isn't a topological invariant). ...
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Explanations of Lebesgue number lemma

From Planetmath: Lebesgue number lemma: For every open cover $\mathcal{U}$ of a compact metric space $X$, there exists a real number $\delta > 0$ such that every open ball in $X$ of radius ...
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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 ...
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Isometry in compact metric spaces

Why is the following true? If $(X,d)$ is a compact metric space and $f: X \rightarrow X$ is non-expansive (i.e $d(f(x),f(y)) \leq d(x,y)$) and surjective then $f$ is an isometry.
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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" ...
14
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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 ...
14
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A separable locally compact metric space is compact iff all of its homeomorphic metric spaces are bounded

The title is a claim my classmate made during our summer vacation :D He showed me a TeX file describing a proof of his claim, and it contains a fairly short but elegant proof. He says that the ...
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Metrics on $\mathbb R^{\mathbb N}$.

I was showing $D(x,y) = \sup_{k \in \mathbb N} \frac{d' (x_k,y_k)}{k}$ induces the product topology on $\mathbb R^{\mathbb N}$. Here $d(x,y)$ is the standard metric on $\mathbb R$ and $d'(x,y) = ...
14
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Convex metric on a contiuum.

I am having troubles with this problem. Translation: A metric $d$ on a continuum is called convex if for any two points $p$ and $q$ of $X$ there is a point $x\in X$ such that ...
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Homeomorphism from square to unit circle

Can we find a homeomorphism from the square $Q_2$ of side length $2$ centered on the origin and the unit circle $S^1$? We can easily define a map $r:Q \longrightarrow S^1$ by $$(x,y) \mapsto ...
13
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What operations is a metric closed under?

Suppose $X$ is a set with a metric $d: X \times X \rightarrow \mathbb{R}$. What "operations" on $d$ will yield a metric in return? By this I mean a wide variety of things. For example, what functions ...
13
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a subclass of quasi metric spaces with properties almost identical to metric spaces

It is well known that passing from metric spaces to quasi-metric spaces (i.e., dropping the requirement that the metric function $d:X\times X\to \mathbb R$ satisfies $d(x,y)=d(y,x)$) carries with it ...
13
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Can one cancel $\mathbb R$ in a bi-Lipschitz embedding?

Let $X$ be a metric space. Suppose that the product $X\times\mathbb R$ admits a bi-Lipschitz embedding into $\mathbb R^{n}$. Does it follow that $X$ admits a bi-Lipschitz embedding into $\mathbb ...
13
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$f\colon I\rightarrow G$ and Gromov $\delta$-hyperbolicity

Please recall that $\left|\int_0^1 f(t)\,dt -w\right|\leq \int_0^1|f(t)-w|\,dt$. In general, let $(X,d)$ be a metric space. Given a function $f:I\to X$ let $m_f\in X$ be such that $d(m_f,w)\leq ...
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Examples of non symmetric distances

It is well known that the symmetric property is $d(x,y)=d(y,x)$ is not necessary in the definition of distance if the triangle inequality is carefully stated. On the other hand there are examples of ...
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Is the intersection of a closed set and a compact set always compact?

I am going through Rudin's Principles of Mathematical Analysis in preparation for the masters exam, and I am seeking clarification on a corollary. Theorem 2.34 states that compact sets in metric ...
12
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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 ...
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Why are quotient metric spaces defined this way?

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 ...
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Continuous images of open sets are Borel?

Consider a Polish space $(X,d)$ and any metric space $(Y,e)$. If we have a continuous surjection $f:X\to Y$ then is the image $f(U)$ of any open subset $U\subset X$ a Borel set in $Y$? I know that ...
12
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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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Every complete, countable metric space has a discrete, dense subset.

Given a complete, countable metric space, say $X$, I'd like to show it has a discrete, dense subset. This seems like an application of the Baire Category Theorem, but that doesn't seem to go anywhere. ...
12
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Converse of a fixed-point theorem

I'm having some trouble furnishing a proof here. Let $(E, d)$ be a metric space such that any $k$-Lipschitz function has a fixed point for $0 < k < 1$. Does it follow, then, that $E$ is ...
11
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What's wrong with this definition of continuity?

Consider this definitions: A function $f:X \to Y$ is continuous at $x\in X$ iff for any open neighborhood $V_{f(x)}$ of $f(x)$ there is an open neighborhood $U_{x}$ of $x$ that gets mapped by $f$ ...
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How to show that $\mathbb R^n$ with the $1$-norm is not isometric to $\mathbb R^n$ with the infinity norm for $n>2$?

Could you please give me a hint to prove that $\mathbb{R}^n$ with the 1-norm $\lvert x\rvert_1=\lvert x_1\rvert+\cdots+\lvert x_n\rvert$ is not isometric to $\mathbb{R}^n$ with the infinity-norm ...
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if every continuous function attains its maximum then the (metric) space is compact

Suppose $(M,d)$ a metric space. I want to show that if every continuous real-valued function on $M$ attains a maximum, then the space must be compact. I was trying to do this by assuming $M$ ...
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Why are metric spaces non-empty?

I'm just second-marking some exam scripts, and I wanted to leap on a question and made the following pedantic remark concerning the model answers: "if the metric space is empty then this proof doesn't ...
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Compactness of the Grassmannian

Let $V$ be a finite-dimensional inner product space. For $0 \leq d \leq \text{dim}(V)$, define the Grassmannian $G(V, d)$ to be the set of all $d$-dimensional linear subspaces of $V$, equipped with ...
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Metric in riemannian Manifold

Let $(M,g)$ be a riemannian Manifold, we can use the metric $g$ to obtain a metric $d_g:M\times M\to \mathbb{R}$. I ask for a kind of converse, we can start with a metric $d:M\times M\to \mathbb{R}$ ...
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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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Is this a metric?

I now that one can show that if $d$ is a metric on a vectorspace $X$ then so is $$\varrho(x,y):=\frac{d(x,y)}{1+d(x,y)}.$$ This easily follows from the fact that the function $s \mapsto \frac{s}{1+s}$ ...
11
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Isometry in compact metric space

Let $(M,d)$ be a compact metric space and $f: M \rightarrow M$ a continuous function. I'm trying to prove that if $d(f(x),f(y)) \geq d(x,y)$ for every $x,y \in M$, then $f$ is an isometry. This is how ...
11
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Show that a metric space has uncountably many disjoint dense subsets if every ball is uncountable.

I was wondering how to show that a metric space $X$ has uncountably many dense subsets $Y_{\alpha}$ such that $Y_{\alpha_1} \cap Y_{\alpha_2} = \emptyset$ if $\alpha_1 \neq \alpha_2$, under the ...