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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$\phi:M\to \mathbb{R}$ continuous, $\phi(x)<\epsilon$ for $x\in X$, then $\phi(x)\le \epsilon$ for $x\in\overline{X}$

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: $\...
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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 ...
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Question about proof ot Tychonoff's theorem for metric spaces

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 ...
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$M\times N$ compact $\implies$ $M$ compact and $N$ compact

I must prove that $M\times N$ compact $\implies$ $M$ compact and $N$ compact using the definition that, if a metric space $M$ is compact, then every cover has an open finite sub cover. $$M=\cup ...
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Hints on showing Cauchy sequence converges

Let $T>0$ and $L\geq0$. Let $C[0,T]$ be the space of all continuous real valued functions on $[0,T]$ with the metric $\rho$ defined by $$\rho(x,y)=\sup_{0\leq t\leq T}e^{-Lt}\left|x(t)-y(t)\right|$...
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How can we fill in some missing details in this proof?

Let $(X,d)$ and $(Y,\rho)$ be metric spaces and $X$ is compact. Suppose $f:X\to Y$ and $f_n:X\to Y$ are continuous functions such that for every $x\in X$, $\rho(f_n(x),f(x))$ decreases to $0$ as $n\...
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2answers
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Closure of sets (specifically regarding the notation)

I'm new to sets and the notation is somewhat confusing to me. I just want to see if what I'm doing makes sense. For the following sets I need determine if it is open, closed, or neither. I also ...
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59 views

For an arbitrary uncountable set of irrational numbers, can I always construct a sequence from them that converge in the rationals?

Suppose you have a set $S$ of uncountably many irrational numbers. Can you construct a sequence of $S$ that converges to a rational number? What I have tried: Since $S$ is uncountable, the inf of ...
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Defining compact sets with closed covers

This question is a continuation of this. My book says that a metric space is compact if and only if: $$M=\cup A_{\lambda}\implies M = A_{\lambda1}\cup\cdots\cup A_{\lambda_n}$$ where each $A_{\...
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2answers
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Could someone please check my proof that $(x_n) \text{ is Cauchy in } X\implies (f(x_n)) \text{ is Cauchy in } Y$

Let $(X,d)$ and $(Y,\rho)$ be metric spaces and $f:X\to Y$. Show that if $f:X\to Y$ is uniformly continuous, then $$(x_n) \text{ is Cauchy in } X\implies (f(x_n)) \text{ is Cauchy in } Y$$ My ...
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20 views

Hyperbolic distance of a point from center in Klein-Beltrami disk model

According to the Wikipedia entry about Klein Beltrami disk, I found that the hyperbolic distance between two points P and Q is determined by the following formula : $$d(P, Q) = \frac{1}{2} \ln \frac{|...
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28 views

compact metric space definition by closed covers

My book says the following: A metric space is compact iff: $$M=\cup A_{\lambda}\implies M = A_{\lambda1}\cup\cdots\cup A_{\lambda_n}$$ where each $A_{\lambda}$ is open. Then, it says that if $A_\...
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45 views

CW complex= cell complex?; compact metrizable spaces

I have a question about finite cell complexes and compact metrizable spaces. In a paper I read the statement: Let $X$ be a compact metrizable space. Then $X$ is a countable inverse limit $\varprojlim\...
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4answers
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prove triangular inequality for $ d(x,y)= \frac{||x-y||}{1+||x-y||}$ [duplicate]

prove triangular inequality for $$ d(x,y)= \frac{||x-y||}{1+||x-y||}$$ that is $d(x,y) \leq d(x,z)+d(z,y)$ ofcourse ||.|| is a norm and has properties of norms this usually works $$ \begin{...
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1answer
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Let $X = C[a,b]$ be a vector space of continuous real functions on the interval $[a,b] \subset{R}$ and $\|x\| = \sup\{|x(t) : t \in [a,b] \}$

Let $X = C[a,b]$ be a vector space of continuous real functions on the interval $[a,b] \subset{R}$ and $\|x\| = \sup\{|x(t) : t \in [a,b] \}$, $x\in X$, norm on $X$. Prove that with $(Ax)t = t^2x(a)$, ...
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52 views

Metric in $\mathbb{S}^1$

Let $\mathbb{S}^1=\{ x=e^{2 \pi ir} | r \in I \}$, if $$d(x,y) = \left\{ \begin{array}{lcc} \min\{s-r,1-s+r\} & \text{if} & r \leq s \\ \\ \min\{r-s,1-r+s\} & ...
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0answers
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Function from space of continuous functions to reals is continuous (Proof Verification)

Question: $C$ is the space of continuous functions from $[0,1]$ to $\mathbb{R}$ under the sup metric. Prove the function $$f:C\to\mathbb{R}\quad f\to \int_0^1 f(t)^2 dt$$ is continuous. My answer: ...
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2answers
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Are these subsets homeomorphic?

Are the two subsets of the Euclidean Plane $[0,1]\times[0,1)$ and $[0,1)\times[0,1)$ homeomorphic or not? My attempt: We need to find a bijective function $f$ from $[0,1]$ to $[0,1)$ such that $f$ ...
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1answer
34 views

Portuguese term for “path metric”

Do anybody knows what is the usual translation to Portuguese for "path metric"? (Given a metric space $(M,d)$, $d$ is called a "path metric" if, given any pair $(x,y)\in M\times M$, there exists a ...
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Proof verificication and question of rigour: $A$, $B$, connected implies union is connected

Don't mark this as duplicate. The other question does not help me figure out how rigorous MY proof is. Problem: Let $A$ and $B$ be connected subsets of a metric space and let $A\cap\overline{B}\neq\...
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Metric space where each continuous function has IVP is connected

The question: Let $X$ be a space such that every continuous function $f:X\rightarrow\mathbb{R} $ does have the following property: if $a<c<b$, $f(x) =a$, and $f(y) =b$, then there exists $z\in ...
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Union of path connected pairwise not disjoint subsets

Problem Let $(X,d)$ be a metric space and let $\mathcal A$ be a family of path connected subsets of $X$ such that for every pair of sets $A,B \in \mathcal A$ there are $A_0,...,A_n \in \mathcal A$ ...
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36 views

existence of certain function on unit interval

I'm trying to solve this exercise in an introductory book on general topology: Let $(X,d)$ be a metric space and $A,B \subset X$ disjoint closed subsets. Show that there exists a continuous function $...
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Is there a complete metric space which has no Cauchy sequence?

Definition: A metric space is said to be complete if every Cauchy sequence is convergent. Now, my question is: Is there a complete metric space which has no Cauchy sequence?
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problem in real analysis about open sets in metric spaces.

For $x = (x_1, x_2, \ldots, x_n)$ and $y = (y_1, y_2, \ldots, y_n)$ in $\mathbb{R}^n$. Let $d_p(x, y) = \Bigg(\sum\limits_{i=1}^n |x_i-y_i|^p\Bigg)^\frac{1}{p}$ for $1 \leq p < \infty$ and $d_\...
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Cauchy sequence in $\mathbb{R}^d$

Is is possible to prove that sequence $\{x_n\}$ with terms in $\mathbb{R}^d$ has limit iff $\forall_{\epsilon > 0}\exists_{N \in \mathbb{N}}\forall_{n,m \ge N} \rho(x_n, x_m) < \epsilon$ as a ...
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1answer
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Is the complement of the closed unit disk in the plane homeomorphic with $\mathbb R^2\setminus \{(0,0)\} $ ? [closed]

Is $\mathbb R^2 \setminus D^2$ , where $D^2=B[0;1]$ is the closed unit disk , homeomorphic with $\mathbb R^2\setminus \{(0,0)\} $ ?
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Is the plane minus a line segment homeomorphic with punctured plane?

Is $\mathbb R^2$ minus a line segment i.e. $\mathbb R^2 \setminus ([0,1]\times \{0\}) $ homeomorphic with a punctured plane $\mathbb R^2\setminus \{(0,0)\}$ ?
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Show that $E \subset \Bbb Q$ is closed in $(\Bbb Q, d)$

Assume $(\Bbb Q, d),$ $d(p, q):= |p -q|$ is a metric space and $E := \{p \in \Bbb Q : 2 < p^2 < 3\} \subset \Bbb Q.$ I have to show that $E$ is closed. I see two ways of proving ...
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35 views

Can an arbitrary metric space be made into a complete and separable metric space?

Can any metric space be made into a separable and complete metric space by suitably choosing an isometry?
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On the matter ; If $f:X \to Y$ is a function with closed graph and compactness preserving then $f$ is continuous

Let $X,Y$ be metric spaces , $f:X \to Y$ be a function , with closed graph , carrying compact sets to compact sets ; then I claim that $f$ is continuous Proof: Let , if possible , $f$ be not ...
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$f$ be a function on real line carrying compact sets to compact sets and fiber of every point under $f$ is closed , is $f$ continuous ?

Let $f:\mathbb R \to \mathbb R$ be a function such that it carries compact sets to compact sets and $f^{-1}(\{x\})$ is closed for every $x \in \mathbb R$ , then is $f$ continuous ? (I know that if $...
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If every five point subset of a metric space can be isometrically embedded in the plane then is it possible for the metric space also?

Let $X$ be a metric space with at least $5$ points such that any five point subset of $X$ can be isometrically embedded in $\mathbb R^2$ , then is it true that $X$ can also be isometrically embedded ...
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$T:\mathbb R^n \to \mathbb R^n $ be an isometry , is $T$ surjective?

Let $T:\mathbb R^n \to \mathbb R^n $ be an isometry and $T(0)=0$ , then $T$ is linear and $T(B[0,1])\subseteq B[0,1]$ so $T:B[0,1]\to B[0,1]$ is an isometry and since $B[0,1]$ is compact so $T|_{B[0,...
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For what functions $f$ is $d_f$ defined by $d_f(x,y) = f(d(x,y))$ also a metric on $X$ assuming $d$ is a metric on $X$

For what functions $f : \mathbb{R}_{\leq 0} \to \mathbb{R}_{\leq 0}$ is it true that for every metric $d$ on a set $X$, the function $d_f$ defined by $d_f(x,y) = f(d(x,y))$ is also a metric on $X$? I ...
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$B$ be an uncountable dimensional real Banach space and $T:B \to B$ be a surjective isometry such that $T(0)=0$ , then is $T$ linear?

Let $B$ be a real Banach space and $T:B \to B$ be a surjective isometry such that $T(0)=0$ , then is $T$ linear ? I know that it is true if $B$ is a Hilbert space ( we only need an inner product ...
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Verification of proof, interior is open.

Let $(\mathbb{X},d_{\mathbb{X}})$ be a metric space and $\emptyset \neq A \subseteq \mathbb{X}$ its subset. Prove that the interior $A^{\circ} =\{ a \in A | \exists\epsilon(a) > 0, B_{\epsilon}^{d_{...
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In $(C[0,1],||.||_{\infty})$ , is the set $\{p(x)\in C[0,1]$, $p(x)$ is a polynomial $:\int_0^1 p(x)dx=1\}$ totally bounded?

Consider the metric space $(C[0,1],||.||_{\infty})$ , in this space , is the set $\{p(x)\in C[0,1]$, $p(x)$ is a polynomial $:\int_0^1 p(x)dx=1\}$ totally bounded ? Please help , Thanks in advance
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Proving that the mapping $ (Ff)(t): C[0, 1] \to C[0, 1] $ is a contraction

This is a follow-up to my previous question. I tried to use John Ma's answer, but couldn't solve this. I need to prove that $F : C[0,1]\to C[0,1]$ is a contraction mapping. $ F $ is defined as ...
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$f$ non-constant on $\mathbb R$ such that for any metric $d$ on $\mathbb R$ , $f:(\mathbb R,d)\to (\mathbb R,d)$ is continuous , is $f$ identity?

Let $f:\mathbb R \to \mathbb R$ be a non-constant function such that for any metric $d$ on $\mathbb R$ , $f:(\mathbb R,d)\to (\mathbb R,d)$ is continuous , then is $f$ the identity function i.e. $f(x)=...
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To characterize uncountable sets on which there exists a metric which makes the space connected

For which uncountable sets $X$ is it true that there exist a metric $d$ on $X$ such that $(X,d)$ is connected ? [ The motivation for this question is : I wanted to characterize function $f : X \to X$...
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Product spaces from metric induced topological spaces $(X,T) \times (X,T)$ into $\Bbb R$ is continuous

Let $(X,d)$ be a metric space and $T$ the induced topology on $X$ by $d$. Prove that the function $d$ from the product space $(X,T) \times (X,T)$ into $\Bbb R$ is continuous. My question is what ...
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$f:S^1 \to \mathbb R$ be continuous , is the set $\{(x,y) \in S^1 \times S^1 : x \ne y , f(x)=f(y)\}$ infinite ?

Let $f:S^1 \to \mathbb R$ be a continuous function , I know that $\exists y \in S^1 : f(y)=f(-y)$ where $y \ne -y $ (since $||y||=1$) , so that the set $A:=\{(x,y) \in S^1 \times S^1 : x \ne y , f(x)=...
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1answer
93 views

Existence of metric $d$ on $\mathbb R$ such that the function $f:(\mathbb R,d) \to (\mathbb R,d)$ ; $f(x)=-x$ is everywhere discontinuous

Does there exist a metric $d$ on $\mathbb R$ such that the function $f:(\mathbb R,d) \to (\mathbb R,d)$ defined as $f(x)=-x$ is everywhere discontinuous ? It is motivated from this question which ...
2
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A set is compact iff every collection… Proof check

I asked this question (A set is compact iff all closed collections of subsets with the f.i.p. have nonempty intersection) a few days ago and was lucky enough to get an answer, but I'm afraid that the ...
2
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1answer
36 views

Infinite Union of Complete Metric Subspaces which is Not Complete

Can anyone think of an example of a metric space $(X,d)$ and an infinite set of complete metric subspaces in $(X,d)$ such that their union is not complete?
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Example Infinite Product of Compact Metric Spaces that is not itself compact [closed]

Can anyone think of an infinite product space of compact metric spaces that us not itself compact?
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Showing this a norm

I want to show that $$\| x \| = \sum_{n=1}^{\infty} \frac{1}{2^n} \frac{\left| x_n \right|}{1+\left| x_n \right|}$$ is a norm. I'm fine showing positivity and the triangle inequality, to show the ...
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1answer
34 views

$\ell_0$ norm and the induced complete metric spaces

I have been reading about the $\ell_0$ norm, wikipedia gives us that "The mathematical definition of the $\ell_0$ norm was established by Banach's Theory of Linear Operations. The space of sequences ...
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14 views

The meaning of “order of congruence” of metric space

I was studying low-distortion embedding of finite metric space, and was confused about the following concept: Order of congruence: A metric space $(X,D)$ has order of congruence at most $m$ if every ...