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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confusion over how to show that f is continuous.

Okay , I'm just going to write down exactly what my book says and then ask my question. "Define the function $f:\mathbb{R}^2\rightarrow \mathbb{R}$ by $f(x_1,x_2)=x_1+x_2$. Prove that $f$ is ...
6
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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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Limits of functions in metric spaces

My teacher said that in the definition of limit, the point in the domain, must be of accumulation, because otherwise the limit is not unique. Why? If the point is isolated, the function is continuous, ...
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0answers
28 views

Show that this f is continuous?

Okay , I'm just going to write down exactly what my book says and then ask my question. "Define the function $f:\mathbb{R}^2\rightarrow \mathbb{R}$ by $f(x_1,x_2)=x_1+x_2$. Prove that $f$ is ...
2
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1answer
18 views

Distance $\Psi(A,B)=\sup_{x\in E}\vert d_A(x)-d_B(x)\vert$ where $d_A(x)=\inf_{y\in A}d(x,y)$.

Let $(X,d)$ be a metric space, assume that $d$ is bounded. Denote $F$ the set of all closed set of $X$. Define $$\Psi(A,B)=\sup_{x\in X}\vert d_A(x)-d_B(x)\vert$$ where $d_A(x)=\inf_{y\in A}d(x,y)$. ...
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1answer
25 views

What would be a standard framework, terminology, or procedure for extending function domains using isomorphisms?

Suppose we have an isomorphism $h:A\rightarrow B$ between spaces $A$ and $B$. Remark: Alternatively, we may consider a isomorphism $h:X(0)\rightarrow X(t)$, where spaces $X(t)$ are parametrized by ...
19
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4answers
2k views

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 ...
2
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1answer
40 views

continuity and closure questions - topology

Let $(X,d)$ be a metric space. Let $U \subseteq (X,d)$. let $k \in (X,d)$. Prove that if $U$ is fixed, $d(U,k)$ is a continuous function of $k$. Prove that $\overline{U} = U \cup V$ where $V$ is the ...
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1answer
28 views

cauchy convergent [on hold]

Q: Let $f$ be a function from a metric space $X$ to metric space $Y$. For any cauchy sequence $\{x_n\}$in $X$ which of the following is true? if $f$ is continuous then $\{f(x_n)\}$ is cauchy ...
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3answers
34 views

existence of unique fixed point

Let $(X,d)$ be a compact metric space and $f:X \to X$ satisfies $d(f(x), f(y))< d(x,y)$ for distinct $x$ and $y$. Then, show that $f$ has a unique fixed point. I tried this question by formulating ...
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1answer
448 views

Picturing Urysohn's Metrization Theorem and Urysohn's lemma?

In my Topological course we have this lemma. [Urysohn's lemma] Suppose that $X$ is a topological space. Then $X$ is normal if and only if, for each pair of disjoint closed subsets $A$ and $B$, there ...
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1answer
17 views

Show that $d(u,v)=\exp(-\max\{j\ge 0, u_k=v_k \space\mbox{for}\space 0\le k\le j\})$ is a distance over $E=\Bbb{R}^\Bbb{N}$.

Let $E=\Bbb{R}^\Bbb{N}$, $u=(u_k)_{k\in\Bbb{N}}$ and $v=(v_k)_{k\in\Bbb{N}}$. Define $$ d(u,v) = \left\{ \begin{array}{ll} \exp(-V(u,v)) & \mbox{if}\quad u\ne v \\ 0 ...
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Is there any standard procedure to properly define a composite metric?

For example, space $A$ has a metric $\rho$, and its subspace $B\subset A$ has a metric $d$, which happens to have much better properties than $\rho$. So if $x_{1},x_{2}\in A\setminus B$, but they are ...
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4answers
52 views

How to give a rigorous proof of this fact about closures of open balls in the euclidean spaces?

Let $n$ be a positive integer, $\vec{a} \in \mathbb{R}^n$, and $r > 0$. Then it is intuitively clear that the closuer of the open ball $$B(\vec{a} ; r) \colon= \{ \vec{x} \in \mathbb{R}^n \colon ...
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2answers
28 views

$C^n[a,b]$ as a normed algebra

I would like to prove that the space of complex valued functions, differentiable $n$ times with continuous derivative, $C^n[a,b]$, with the metric defined by the norm ...
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1answer
25 views

Is my reasoning accurate?

$$\text{d}_{H}(A,B) = \max\left\{ \sup_{a\in A} \inf_{b\in B} \text{d}(a,b),\sup_{b\in B} \inf_{a\in A}\text{d}(a,b)\right\}$$ where $A$ and $B$ are two closed subsets of a metric space $(E,d)$ is a ...
2
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2answers
37 views

Let $(Y,\rho)$ be a metric space and $\rho : Y \times Y \rightarrow \mathbb{R}$ Prove that $\rho$ is a continuous function on $Y \times Y$.

Let $(Y,d)$ be a metric space and $d : Y \times Y \rightarrow \mathbb{R}$ Prove that $d$ is a continuous function on $Y \times Y$. I was thinking of the following : If $(a_{1},a_{2}) \in Y \times ...
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3answers
43 views

Showing $f$ is continuous on $M$ if $M=\bigcup_{n=1}^{\infty} U_n$

Let $f:(M,d)\to (N,\rho )$. If $M=\bigcup_{n=1}^{\infty} U_n$, where each $U_n$ is open, and if $f$ is continuous on each $U_n$, show that $f$ is continuous on $M$. Attempt: I note that ...
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2answers
53 views

Prove that this is a metric space?

I'm supposed to show that If X is the set of all functions on the interval $[a,b]$ and $\displaystyle d(f,g)= \int^{b}_{a}|f(x)-g(x)|dx\,$, then $(X, d)$ is a metric space. But I don't think it ...
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0answers
128 views
+50

TQFTs and Feynman motives

Is a topological quantum field theory metrizable? or else a tqft coming from a subfactor? For a given metric, are there always renormalization and Feynman diagrams? Is there always a Feynman motive ...
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1answer
27 views

On Pseudometric

How a pseudometrics induces topology? Can anyone discuss on this topic or give any good reference?
2
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1answer
19 views

Is $d(i,j) = 1-\textrm{corr}(i,j)$ a metric?

I need to make sure that this function is a metric: $d(i,j) = 1-\textrm{corr}(i,j)$ where $\textrm{corr}(x,y)$ is the Pearson correlation coefficient which ranges from $[-1,1]$. With this scaling I ...
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1answer
31 views

proof that a set of all bounded real valued functions is complete.

I am trying to understand the proof below. I know that a set A is complete if all Cauchy sequences converges in A. I don't understand 7th line of the proof. Why do we consider particular $x_0 \in X$ ...
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0answers
16 views

vectors representation using matrix. [on hold]

please any one can help with following attached questions? Thank you.
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1answer
29 views

Constructing a countable dense subset of a totally bounded set

Given a metric space $(X,d)$, and (non-empty) totally bounded set $E$ in $X$, is it possible to construct $D \subseteq E$ which is countable and dense? I feel that this should definitely be possible. ...
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1answer
104 views

Is there always an equivalent metric which is not complete?

I have seen that completeness is not a topological property like compactness or connectedness. I have seen some examples also showing that there are two equivalent metrics one of which is complete and ...
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1answer
34 views

$\partial(S') \subset \partial S$ iff $S' \cap S^o \subset (S')^o$

Usually I can come up with some ideas but this time I don't. It would be great if you can tell me how I would make use of the first part of the question to prove the equivalent relation. Question: ...
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2answers
139 views

Critique this proof on compactness.

Problem: Prove or disprove, the metric space $X$ containing infinitely many points with the discrete metric is compact. Write a proof in the language of sequences and covers Proof: Take $(1/n) \to ...
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1answer
37 views

Definition of a metric space: why $E\times E\rightarrow\mathbb{R}$?

In the definition of a metric space Let $E$ be a set and $d:E\times E\rightarrow\mathbb{R}$ be a function. $d$ is a distance on $E$ if ..., why is the function $d:E\times ...
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1answer
19 views

Distance to a closed set is continuous.

I want to prove that given a metric space $(M,d)$ and $F \subset M$, then the function $f_F: M \to \Bbb R$ given by $f_F(x) = d(x,F) = \inf\{d(x,y) \ : \ y \in M\}$ is continuous. Take $x \in M$. If ...
2
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1answer
17 views

Extend Metric Space Challenge

Let $(E, D)$ be a metric space. Consider $D_1: E\times E \to \mathbb{R}$ where $$ D_1(x,y)=\frac{D(x,y)}{1+ D(x,y)}. $$ I read some note about it but I want to find why $D_1$ is also a metric and ...
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2answers
24 views

Basic Topology - Metric Spaces [closed]

$\textbf{Problem}$: Check that $d(f,g)$ = $max_{a\leq x\leq b}$ $|f(x)-g(x)|$ defines a metric on $C([a,b])$, the collection of all continuous and real valued functions defined on the closed interval ...
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1answer
13 views

Subsets of a metric space in which Hausdorff semi-distance is symmetric

These are the definition of Hausdorff distance and Hausdorff semi-distance for subsets of a metric space $X$. ‎‎Hausdorff semi-distance of two subsets ‎$‎A‎, B‎ \subset X$ is defined as below: ‎$‎d(A ...
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2answers
33 views

if $A$ is open in $M$ and $B$ is open in $N$, then $A \times B$ is open in $M \times N$

where $d((m_1,n_1),(m_2,n_2)) = d_M(m_1,m_2) + d_N(n_1,n_2)$ By some propositions, $A$ is open in $M$ if there exist an open set $K_1$ such that $A = M \cap K_1$ Also, there exist an open set $K_2$ ...
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1answer
27 views

Cauchy sequence and metrics

I'm having trouble with another analysis homework problem: Let $x_n$ be a sequence in $\mathbb{R}$ such that $d(x_n, x_{n+1}) \le \frac{d(x_{n-1},x_n)}{2}$. Show that $x_n$ is a Cauchy sequence. I ...
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1answer
32 views

Convergence and metric - Proof?

Let $(x_n)$, $(y_n)$ be two sequences in a metric space $(P,d)$. Suppose $(x_n)$ converges to $x$ and $(y_n)$ converges to $y$. Prove that $\displaystyle\lim_{n \to \infty} d(x_n,y_n) = d(x,y)$ My ...
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2answers
24 views

Sequential Equivalence Implies Topological Equivalence

Define two metric spaces $(M,d)$ and $(M,\rho)$ to be equivalent, denoted $d\sim p$, to mean that: Topological Definition $\forall x\in M: \forall \epsilon>0 \exists \delta_1>0, \delta_2>0: ...
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1answer
275 views

Non-empty intersection of open balls in $R^n$ contain open balls

I want to prove that if the intersection of two open balls about the points $x, y$ (resp.) is non-empty, then there exists a third ball centered at some point $z\in B_{\epsilon 1}(x)\cap B_{\epsilon ...
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2answers
395 views

Sequence has a convergent subsequence in R^n

Suppose A is a closed and bounded subset of R^n. Let {ak} be a sequence in A. Thus, the elements of {ak} are: (a11,a12,...,a1n), (a21,a22,...,a2n), ... ... (ak1,ak2,...,akn), ... We are not sure if ...
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1answer
37 views

Prove the following is a metric space…

I need to prove the following is a metric space over the integers: $b \geq 2$. For distinct integers $x, y$. Let $N(x,y)$ be the greatest integer $n$ such that $b^n$ divides $(x - y)$. Let $d(x,y) = ...
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2answers
23 views

What does this function converge to in $\mathbb{R}$ equipped with discrete metric?

We're given this function $f_n (x) = \begin{cases} 0 \ \mbox{ if $x <1/n$}\\ 1 \ \mbox{ if $x \geq 1/n$} \end{cases}$ I think it converges pointwise to $f(x) = \begin{cases} 0 \ \mbox{ if $x ...
2
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1answer
32 views

Pointwise convergence implies uniform convergence in discrete metric space $(X,d)$?

In general, does this hold for a sequence of functions in an arbitrary $X$? For a sequence to converge in the discrete metric, the sequence needs to become a constant sequence for a sufficiently large ...
2
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1answer
30 views

Convergence of sequence and interior points

For a subset $A \subseteq X$, consider the statement, "$x$ is an interior point of $A$ iff for every sequence $(x_m)$ in $X$ converging to $x$ there exists $n \in \mathbb{N}$ such that for all $m > ...
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2answers
37 views

Open set in subspace not open in the entire space example

I am stuck with the following problem: X is a metric space. Suppose that Y is a subspace of X. Give an example that an open set in Y is not open in X. My own approach was this: Suppose U is a subset ...
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2answers
25 views

Convergence and finer topology

Can convergent of sequence be used to determine which topology is finer(in general topological space). I am asking this is question in effect of theorem on metric space: 'topology 1 is finer than ...
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1answer
33 views

Connected Sets on Metric Spaces

I'm taking a first course in real analysis, and we're using Rudin's Principles of Mathematical Analysis as our main (only) book. In chapter two, Rudin discusses basic topology from the point of view ...
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1answer
36 views

Distance between a point and a closed set in metric space

Here is what I am thinking. Let (X,d) be a metric space and let C be a closed subset of X. Fix any poin p in X. Then, there exists a point q in C such that d(p,q) = distance(p,C). I think this ...
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1answer
598 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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1answer
47 views

Example of a locally compact metric space whose completion is not locally compact

Can someone suggest an example of a locally compact metric space whose completion is not locally compact?
2
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1answer
28 views

What are the epis in Met?

I have an assignment to precisely describe epimorphisms and monomorphisms in Met (category whose objects are Metric spaces and whose morphisms are contractions). I have shown that Mono $\iff$ ...