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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Continuity of the metric function

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) - ...
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If $A$ is compact and $B$ is closed, show $d(A,B)$ is achieved

Let $A, B$ be subsets of a metric space $X$. If $A$ is compact and $B$ is closed, show that the distance between $A$ and $B$ is achieved. Attempt at a proof: Let $A$ be compact and $B$ be ...
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Why doesn't $d(x_n,x_{n+1})\rightarrow 0$ as $n\rightarrow\infty$ imply ${x_n}$ is Cauchy?

What is an example of a sequence in $\mathbb R$ with this property that is not Cauchy?
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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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A isometric map in metric space is surjective? [duplicate]

Possible Duplicate: Isometries of $\mathbb{R}^n$ Let $X$ be a compact metric space and $f$ be an isometric map from $X$ to $X$. Prove $f$ is a surjective map.
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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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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 ...
15
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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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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 ...
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Prove the map has a fixed point

Assume $K$ is a compact metric space with metric $\rho$ and $A$ is a map from $K$ to $K$,$\rho (Ax,Ay) < \rho(x,y)$ for $x\neq y$.Prove A have a unique fixed point in K. The uniqueness is easy.My ...
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which of the following metric spaces are complete?

Which of the following metric spaces are complete? $X_1=(0,1), d(x,y)=|\tan x-\tan y|$ $X_2=[0,1], d(x,y)=\frac{|x-y|}{1+|x-y|}$ $X_3=\mathbb{Q}, d(x,y)=1\forall x\neq y$ $X_4=\mathbb{R}, ...
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An open ball is an open set

Prove that for any $x_0 \in X$ and any $r>0$, the open ball $B_r(x_o)$ is open. My attempt: Let $y\in B_r(x_0)$. By definition, $d(y,x_0)<r$. I want to show there exists an ...
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Metric Spaces Analysis

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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preservation of completeness under homeomorphism

Does homeomorphic metric spaces preserves completeness?I mean two metric space which are homeomorphic and one of them is complete$\Rightarrow$ another one is also complete?
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Let $L_p$ be the complete, separable space with $p>0$.

Let $L_p$ be the complete, separable space with $p>0$. $\mathbf{J}=\{I = (r,s] \}$ where $r$ and $s$ are rational numbers. $\mathbf{A}$ is the algebra generated by $\mathbf{J}$, with ...
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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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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 ...
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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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Sum of Cauchy Sequences Cauchy?

Let $(X,+)$ be an abelian group and $d$ a metric on $X$. Suppose $\{a_n\}$ and $\{b_n\}$ are Cauchy sequences. What conditions on the relation between the group operation and the metric are sufficient ...
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Notions of equivalent metrics

Let $X$ be a set, and $d,d'$ two metrics on $X$. Consider the identity map $i : (X,d) \to (X,d')$ as a map of metric spaces. There are (at least) three reasonable notions of equivalence for $d$ and ...
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Finding a homeomorphism $\mathbb{R} \times S^1 \to \mathbb{R}^2 \setminus \{(0,0)\}$

Are there any specific 'tricks' or 'techniques' in finding homeomorphisms between topological or metric spaces? I'm trying to construct a homeomorphism between $\mathbb{R} \times S^1 \to \mathbb{R}^2 ...
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Irrational P-adics

$\mathbb{Q}_p$ is completion of $\mathbb{Q}$ by defining a new metric. So, with respect to this new metric they are complete. I just want to be sure, are there p-adic rationals? If there are P-adic ...
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Show that $d$ is a metric on $\mathbb C^n$

On $\mathbb C^n$, define $||z||=(\sum_{j=1}^{n}|z_j|^2)^\frac{1}{2}$ and for $x,z\in\mathbb C^n$ define $d(z,w)=||z-w||.$ Prove that $d$ is a metric on $\mathbb C^n$. My attempt: I need to show ...
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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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If $X$ is a connected subset of a connected space $M$ then the complement of a component of $M \setminus X$ is connected

I have an exercise found on a list but I didn't know how to proceed. Please, any tips? Let $X$ be a connected subset of a connected metric space $M$. Show that for each connected component $C$ of ...
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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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What are some motivating examples of exotic metrizable spaces

Among topological spaces, the metric spaces are usually considered to be the tame animals. Describing the topological notion of closeness by a distance is so intuitive (as opposed to the abstract ...
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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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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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Proof of the Lebesgue number lemma

I want to prove the Lebesgue number lemma: Let $(X, d)$ be a compact metric space. Then given an open cover $\mathcal{A}$ of $X$, there exists $\delta \gt 0$ such that for each subset of $X$ ...
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Examples of function sequences in C[0,1] that are Cauchy but not convergent

To better train my intuition, what are some illustrative examples of function sequences in C[0,1] that are Cauchy but do not converge under the integral norm?
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The space of continuous, bounded functions from a metric space $X$ to $\mathbb R$

Let $(X,d)$ be a metric space. We denote by $C_b(X;\mathbb{R})$ the space of continuous and bounded functions from $X$ into $\mathbb{R}$, equipped with the sup-norm metric. We define a mapping $O: X ...
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Continuous extension of a uniformly continuous function from a dense subset.

I'm trying to understand an alternative proof of the idea that if $E$ is a dense subset of a metric space $X$, and $f\colon E\to\mathbb{R}$ is uniformly continuous, then $f$ has a uniform continuous ...
4
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Do projections onto convex sets always decrease distances?

Suppose $(M, d)$ is some $\ell_p$ metric space (not necessarily Euclidean), and $C \subseteq M$ is a closed convex set. Consider the projection function $f_C:M\rightarrow C$ defined such that: ...
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Is Completeness intrinsic to a space?

Is completeness an intrinsic property of a space that is independent of metric? For example, since $\mathbb{R}^n$ is complete with the Euclidean metric, is it complete with any other metric? If ...
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For $F$ closed in a metric space $(X,d)$, is the map $d(x,F) = \inf\limits_{y \in F} d(x,y)$ continuous? [duplicate]

Possible Duplicate: Continuity of the metric function For $F$ closed in a metric space $(X,d)$, is the map $d(x,F) = \inf\limits_{y \in F} d(x,y)$ continuous? I think it is, but I'm having ...
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Continuous functions between metric spaces are equal if they are equal on a dense subset

If two functions defined on metric spaces $X$ and $Y$ are equal on a dense subset of $X$ and are continuous also, then are they equal on all of the metric space $X$?
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What about the continuity of these functions in the uniform topology?

Let $f$, $g$, $h \colon \mathbf{R} \to \mathbf{R}^\omega$ be defined by $$\begin{align*} f(t)&:=(t,2t,3t,\ldots),\\\\ g(t)&:=(t,t,t,\ldots),\\\\ ...
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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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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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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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M compact $p\in M$ , there exist $f:M-p\to M-p$ continuous bijection but not homeomorphism?

Let M be a compact metric space. We know that if $ g:M\to M$ is a continuous bijection then it's a homeomorphism. But I want to know, if I have a continuous bijection $ f:M - \left\{ p \right\} \to M ...
8
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280 views

pointwise limit on a complete metric space

Let $\{f_n: X\rightarrow \mathbb{R}\}$ be a sequence of continuous real-valued functions on a complete metric space, $X$. Suppose this sequence has a pointwise limit, $f$. How easy is it to see that ...
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Continuity of $d(x,A)$

I am doing a head-check here. I keep seeing this theorem quoted as requiring $A$ to be closed (as in Is the function distance continuous?), but I don't think that it is needed. Theorem. Let ...
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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 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 ...
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Prokhorov metric vs. total variation norm

Let $(S,d)$ be a metric space and let $\mathcal P(S)$ denote the space of Borel probability measures on $S$ endowed with the Prokhorov metric $\pi:\mathcal P(S)\times \mathcal P(S)\to \mathbb R_+$ ...
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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 have problem imagine how this space looks like. I think it contains of all sequences containing ones and ...