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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When is the completion of a topological vector space a Frechet space?

Suppose $X$ is a topological vector space with the metric topology. If we take the completion of $X$ with respect to the metric, will we get a Frechet space? Are there any extra conditions needed to ...
6
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1answer
26 views

Let $A$ be any subset of $\mathbb R^{+}$ , then there exist a metric space $(X,d)$ such that $d:X \times X \to A \cup \{0\}$ is a surjection?

Let $A$ be any subset of the set of positive real numbers $\mathbb{R}_+$ ; then does there exist a metric space $(X,d)$ such that $d\colon X \times X \to A\cup\{0\}$ is a surjection ?
13
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1answer
380 views

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 ...
3
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1answer
16 views

Open set in Hilbert Cube.

Any open set in the Hilbert Cube is the union of open subsets of the form $$U_1 \times ... \times U_n \times X_{n+1} \times .... \times X_{n+k} \times...$$ where $X_k := [0, \frac{1}{k}]$ for $k \in ...
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33 views

The triangle inequality for shortest paths of graphs

In why-the-triangle-inequality I found the statement: for example if $d(a,b)$ measures the "length" of the "shortest path" between points $a$ and $b$ (and this can be interpreted quite ...
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3answers
66 views

Exercise on Metric space

I hve this exercise it is very simple but i don't know how to write the answer Let $A$ be a nonempty set in $(E,d)$, for $\varepsilon>0$ we note $$V_{\varepsilon}(A)=\{x\in E, ...
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16 views

Equivalence of Definitions of completion of metric space

I've come across two different definitions for a completion of a metric space and am trying to figure out why they are equivalent. The definitions are: 1) Let $(X,d)$ be a metric space. Then ...
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21 views

Rationals in an interval $[a,b] \in \Bbb R$

(i) For which real values $a$ and $b$, ($a < b$), is the set $[a,b] \cap \Bbb Q$ open in $(\Bbb Q, d)$, (where $d(x,y)= \lvert x-y \rvert$)? (ii)For which real values $a,b$ is the set $[a,b] \cap ...
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1answer
31 views

Statiscal Distance Properties

Please anyone could give me any idea of how prove the following property of statistical distance: $d(AB,CD)\leq d(A,C)+d(B,D)$ Remenber that: $(X,d)$---> Metric Space $d:X\times X\rightarrow ...
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2answers
44 views

for which values of $x,y$ is $[x,y]\cap \mathbb{Q}$ closed?

for which values of $x,y$ is $[x,y]\cap \mathbb{Q}$ closed in the metric space $(\mathbb{Q},d)$ where $d(x,y) = |x-y|$ my attempt: I suspected it's closed for all real numbers: let $x,y \in ...
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1answer
36 views

Question in analysis: subset of open interval in $\Bbb R$

Consider metric space $(X,d)$, $X=(a,b)\subset \Bbb R$, $d(x,y)= \lvert x-y \rvert$. Let a subset $S \subset (a,b)$ be open and closed. Show that either $S=(a,b)$ or $S= \emptyset$. There's a ...
14
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2answers
230 views

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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1answer
28 views

Prove that $(0,1)\times (0,1)$ is open in $\mathbb R^2$. [on hold]

Consider a plane $\mathbb R^2$ with the metric $$d(x, y) = \sqrt{|x_1 - x_2|^2 + |y_1 - y_2|^2}.$$ Show that $U = (0,1) \times (0,1)$ is an open set in $\mathbb R^2$ under this metric. How to ...
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1answer
39 views

Determine if a function is a metric

I have been asked the following question in one of my tests. I'm not sure of how to do it. Consider the plane $X = \Bbb R^2$. For each of the following two proposed distance functions, determine ...
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3answers
439 views

If Y is a metric subspace of X. A subset E of Y is open relative to Y if and only if E=Y ∩ G for some open subset G of X.

If we take X=R and Y=Z and E=N i.e. {1,2,3,4,5.........} then since for this case E is open in Y (as Y is itself an entire metric space) however there does not exist any open set G in X for this ...
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1answer
16 views

Non-Lipschitz homeomorphism from compact metric space to itself

Is it possible to find a compact metric space $(X,d)$ with more than one point and a homeomorphism $\varphi:(X,\tau) \to (X,\tau)$ where $\tau$ is the topology induced by $d$ such that $$(\forall N\in ...
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2answers
19 views

The infimum $\inf_{(a,b) \in A\times B} \; \rho(a,b)$ is attained for any two compact sets $A,B$

Let $A,B$ be compact sets in $(S,\rho)$. Define $\rho(A,B)$ by $$\rho(A,B) = \inf_{(a,b) \in A\times B} \; \rho(a,b)$$ Show that there exists $a_0 \in A, b_0 \in B$ s.t. $$\rho(A,B) = \rho(a_0,b_0)$$ ...
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2answers
33 views

Direct sum of metrizable spaces.

I managed to prove that an arbitrary direct sum of metrizable spaces is again metrizable. However, I used the theorem that says that a hausdorff regular space is metrizable if and only if there existd ...
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2answers
28 views

prove of topology and metric spaces [on hold]

Prove or disprove $f: A \to B$ a function from $A$ to $B$. $A_i$ subset of $A$ and $B_i$ subset of $B$. If $A_0 \subset A_1$ then $f(A_0) \subset f(A_1)$ $f(A_0 \cup A_1) = f(A_0) \cup f(A_1)$ ...
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2answers
41 views

Let $a$ and $b$ be arbitrary real numbers with $a < b$. Show that $[a,b]$ is closed by proving its complement is open.

Let $a$ and $b$ be arbitrary real numbers with $a < b$. Show that $[a,b]$ is closed by proving its complement is open. I don't have any idea on this, can anyone help me on this?
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0answers
47 views

Help with general topology questions [on hold]

Given $P_0=(x_0,y_0)$ and $P_1=(x_1,y_1)$ points in $\mathbb{R}^2$, define the distance between $P_0$ and $P_1$ as $$d(P_0,P_1)=\sqrt{(x_0-x_1)^2+(y_0-y_1)^2}.$$ In $\mathbb{R}^2$, the equivalent of ...
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1answer
38 views

Question about a topology proof [on hold]

Hi. I need help with this simple question. I am not able to get this one.
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20 views

Let $Q \subset (m,n)$ be a subset which open and closed, show that $ Q = (m,n)$ or $ Q = \emptyset$

Consider the metric space $((m,n),d)$ where $(m,n) \subset \mathbb{R}$ and $d(x,y) = |x-y|$ Let $Q \subset (m,n)$ be a subset which open and closed, show that$ Q = (m,n)$ or $ Q = \emptyset$ there ...
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1answer
108 views

shortest path in complete metric space

Let $(X,d)$ be a complete connected by arcs metric space. We define the length of a continuous path $\gamma: [0,1] \rightarrow X$ to be \begin{equation*} \sup\limits_{0=a_{0}<a_{1}<... a_{n}=1} ...
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3answers
40 views

doubt with proof in genral topology [on hold]

let Z and Q represent the integers and the rationals, respectively. prove that Z is a closed subset of R. Frankly I don't have an idea how to start. Can anyone please help me with this proof.
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1answer
12 views

showing restricted metric still forms a complete metric space

Let $(A,d)$ be complete. Let $B$ be a closed subset of $A$. Then show the metric space $(B,d|_{B\times B})$ is complete. I have shown from a theorem that $(B,d)$ is complete, but I am not sure how to ...
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3answers
52 views

Proof of questions with general topology. [on hold]

Let $A$ be any subset of $\Bbb R$ with $|A| < \infty$. Prove that $A$ is closed. Can anyone please help me with this proof?
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1answer
46 views

Prove that, for $x \in \mathbb R$ and $\delta_x > 0$, the open interval $(x-\delta_x, x+\delta_x)$ is itself an open set [duplicate]

Prove that, for $x \in \mathbb R$ and $\delta_x > 0$, the open interval $(x-\delta_x, x+\delta_x)$ is itself an open set. I am preparing for my exam and we will be asked to prove various ...
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2answers
53 views

In a metric space we have $B(x,r) = B(y, s)$, is it necessary that $x = y$ and $r = s$??

If in a metric space we have $B(x,r) = B(y, s)$, is it necessary that $x = y$ and $r = s$? I think that the center of the balls i.e. $x$ and $y$ must be same but the radius $r$ and $s$ may not be ...
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0answers
18 views

Entropy of isometric extension

A similar question to mine was asked before at the address below but it was not answered there so I am asking it again. Also there is a more specific case I am interested in. Topological entropy of ...
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2answers
46 views

Show that $d_V$ is a metric

Problem: For points $p = (p_1, p_2)$ and $q = (q_1, q_2)$ in $\mathbb{R}^2$ define: $d_V(p,q) = \begin{cases}1 & p_1\neq q_1 \ or\ |p_2 - q_2|\geq 1 \\ |p_2 - q_2| & p_1= q_1 \ and\ |p_2 ...
3
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2answers
93 views

Construct a set of real numbers whose limit points comprise the set of integers $\mathbb{Z}$

My thought process is the following: Let $S=\{ m + \frac{1}{n}| m \in \mathbb{Z},n \in N \}$. Then I need to show that the limit points of $S$ are indeed the integers and that these are the only ...
21
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2answers
528 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 ...
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1answer
35 views

How to show that this set is closed in $\mathbb{R}^n$?

For an open set $\Omega\subseteq\mathbb{R}^n$, let $K_j$ be the set of points $x$ of $\Omega$ such that $\text{dist}(x,\partial\Omega)\geq1/j$ and $|x|\leq j$. Question : Why is $K_j$ closed ? ...
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0answers
49 views

Product spaces $X = Y = \mathbb R$

Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces. Let $d_{X \times Y} : X \times Y \rightarrow \mathbb R_+$ be given by $$d_{X \times Y}((x_1,y_1),(x_2,y_2)) := d_X(x_1, x_2) + d_Y(y_1, y_2)$$ How ...
2
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4answers
47 views

Proving $d_1$ is a metric.

If $d$ is a metric on a set X, then $d_1 = \frac{d(x,y)}{1+d(x,y)}$ is also a metric. I have proved the other conditions of being a metric except the triangle inequality. Please help!!
27
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470 views

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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1answer
21 views

Sequential continuity on metric spaces

Please give me a hint for proving this statement: Let $(X,d)$ and $(Y,d')$ be metric spaces, $f$ a function from $X$ to $Y$. If $f^{-1}(B) $ is closed in $X$ for all closed subset $B$ of $Y$, then ...
6
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1answer
174 views

non-symmetric version of compact = totally bounded + complete

It is well-known that a metric space is compact iff it is totally bounded and complete. More generally, it is well-known that a uniform space is compact iff it is totally bounded and complete. Is ...
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1answer
34 views

Computing $\sup_{\left \| u \right \|=1} d(u,F)$ in a closed subspace of a normed space.

I have come across with the following problem: Let $E$ be a normed space and $F$ a closed subspace of $E$. It's asking to compute $\sup\limits_{\left \| u \right \|=1} d(u,F)$. What I know it's ...
2
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0answers
23 views

Problems in metric space including matrices. [closed]

Let $M(n, \Bbb R)$ denote the set of a real $n \times n$ matrices. We can always define a linear isomorphism between $M(n, \Bbb R)$ and $\Bbb R^{n^2}$....where the isomorphism is defined as for any ...
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1answer
58 views

When does a homeomorphism preserve Cauchy Sequence?

Is there any necessary and sufficient condition under which a the inverse function in a homeomorphism will preserve Cauchy Sequence ?
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1answer
36 views

Separability of the Set of Bounded Functions over [0,1]

I'm working through Neal Carothers' Real Analysis and I'm stuck on trying to show that the set $B$ of bounded, real-valued functions over $[0,1]$ is not separable. The metric of this set is ...
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0answers
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How to prove a constructed set is a (n,ϵ)-spanning set for a [0,1] -> [0,1] homeomorphism

More specifically, I'm trying to figure out how to show that the following set is an $(n,ϵ)$-spanning set: $S = \{f^{-i}\big(\frac{j}{N}\big) \big| i = 0,1...n-1, j=0,1,...N\}$ where $N$ is selected ...
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2answers
19 views

Prove this function is lower semi-continuous

Let $X$ be a metric space, and $B$ his borel $\sigma$-algebra. Fix $r>0$ Let $\mu$ be a probability measure on $(X,B)$ and define $f(x)=\mu(B(x,r))$. Show that $f$ is lower semi continuous. What ...
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1answer
17 views

Is there a Heine cretierion of liminf of a function?

Lately i've been struggling with understanding the meaning of $\liminf_{x\to x_0}f(x)$ assuming $f:X\to\mathbb C$ for $X$ a metric space, or for that matter $f:\mathbb R\to \mathbb R$. Could you give ...
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0answers
22 views

Connectedness ( cardinality and connectedness) [duplicate]

$(X,d)$ metric space and $A\subset X$ and $A$ is connected. $$ \text{Card}(A) > 2 \implies \text{Card}(A) \geq \text{Card}(\mathbb{R}).$$ How do I prove it ?Waiting for your help?
0
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1answer
21 views

What is an example of a connected subset of $\mathbb{R}^2$ where the interior is not connected?

In $\mathbb{R}^2$ with the usual metric, could this be an open disk, e.g. $ \{(x,y) : x^2 + y^2 \le 2\}$? Thanks in advance!
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2answers
42 views

confusion over Finite intersection property

It is stated that $A_n={(\frac{-1}{n},\frac{1}{n})}$, then arbitrary intersection of open sets need not be open is true as in this case $\bigcap_{i=1}^{\infty}=\left \{0 \right \}$ is not open. Now ...
2
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1answer
30 views

Net convergence in metric spaces

This is a question about convergence of nets which I don't quite understand yet. In metric spaces convergence of sequences encodes the topology but suppose we want to study convergence of nets even ...