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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If $X$ is a metric space such that any metric space $Y$ , which is a homeomorphic image of $X$ , is complete , then is $X$ compact? [duplicate]

Let $X$ be a compact metric space , then it is easy to show that every homeomorphic image metric space of $X$ is complete . Is the reverse true ? That is if $X$ is a metric space such that any ...
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19 views

Let $V$ be a NLS (over $\mathbb R$ ) of dimension $>1$, then for any $0 \le a < b$ , is the set $\{v \in V : a < ||v|| < b\}$ connected?

Let $V$ be a normed linear space (over $\mathbb R$ ) , then for any $0 \le a < b$ , is the set $\{v \in V : a < ||v|| < b\}$ connected ? I know that if $V$ is the space of complex numbers ...
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3answers
76 views

Is $\{x : x\sin{\frac{1}{x}} = 0 \}$ closed in $\mathbb{R}$?

My professor says that $0$ is the only limit point of this set, and $0$ is in this set since $\sin$ is bounded between $-1$ and $1$, and it oscillates between these values so you can multiply ...
2
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1answer
37 views

To show that the annulus $\{x \in \Bbb R^2 : 1 < ||x|| < 2\}$ is connected.

To show that the annulus $\{x \in \Bbb R^2 : 1 < ||x|| < 2\}$ is connected. I want to do it without path-connectedness or polygon-connectedness using the fact continuous image of a connected ...
2
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1answer
46 views

Is $x \in X$ isolated iff the ideal $M_x := \{f \in C(X, \mathbb R) :f(x)=0\}$ of $C(X, \mathbb R)$ is principal ?

Let $(X,d)$ be a metric space , then is $x \in X$ isolated iff the ideal $M_x := \{f \in C(X, \mathbb R) :f(x)=0\}$ of $C(X, \mathbb R)$ is principal ? Do we need completeness of $X$ ?
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1answer
22 views

If $(X,d)$ is a finite metric space , then is every prime ideal of $C(X, \mathbb R)$ maximal?

If $(X,d)$ is a finite metric space , then is every prime ideal of $C(X, \mathbb R)$ maximal ? The thing is , since $X$ is finite , so it is compact , so ideal $M$ is maximal iff it is of the form ...
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72 views

Is $\{x : \sin{\frac{1}{x}} > 0 \}$ open/closed in $\mathbb{R}$?

The set consists of elements that satisfy $0 < \frac{1}{x} < \pi$ (and $2\pi$ repetitions of these solutions for $x$) but I'm having a difficult time visualizing any open balls around any points ...
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1answer
24 views

How to get tietze extension theorm (for metric spaces) with arbitrary co-domain of real valued function

I know the tietze extension theorem on with bounded range namely " If $F$ is a closed subset of a metric space $X$ such that $f:F \to [a,b]$ is a real valued continuous function , then there is a ...
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2answers
78 views

Is $\{x : \sin{\frac{1}{x}} = 0 \}$ open in $\mathbb{R}$?

The set consists of elements that satisfy $\frac{1}{x} = n\pi$ (or $x = \frac{1}{n\pi}$), but I can't visualize any open balls around any points because this is a trigonometric function in ...
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2answers
16 views

Definitions of cluster and condensation points

Pugh in Real Mathematical Analysis defines $p$ as a cluster point of $S$ if each $M_rp$ (r-neighborhood of $p$) contains infinitely many points of $S$. He defines $p$ as a condensation point if each ...
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1answer
18 views

Proposition on a locally lipschitz mapping in the first variable

Suppose $f: M_{1} \times M_{2} \to M_{3}$ is a locally Lipschitz continuous mapping in the first variable between the product metric space $M_{1} \times M_{2}$ and the metric space $M_{3}$, in the ...
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0answers
26 views

Abstract characterization of $R$ or $R^n$ as a metric or topological space?

Given a metric space $M$, are there metric space properties (path connected, second countable, etc.) that force $M$ to be isomorphic to some Euclidean space?
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18 views

A question concerning to show that $V(I)$ is open if $I$ is radical ideal

Let $I:=(f_1 ,...,f_k)$ be a finitely generated ideal of $C(X,\mathbb R)$ such that $\mathrm{rad}(I)=I$, $f:=\sqrt{\sum_{m=1}^k |f_m|}=\sum _{m=1}^k g_mf_m$ where $g_i $'s are real valued ...
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1answer
35 views

$(M,d)$ is a compact metric space and $f:M \to M$ is bijective such that $d(f(x),f(y)) \le d(x,y) , \forall x,y \in M$ , then is $f$ an isometry?

$(M,d)$ is a compact metric space and $f:M \to M$ is an bijective function such that $d(f(x),f(y)) \le d(x,y) , \forall x,y \in M$ , then is $f$ an isometry i.e. $d(f(x),f(y)) = d(x,y) , \forall x,y ...
2
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1answer
81 views

If $f:\mathbb R \to \mathbb R$ is an additive function whose graph is $G_{\delta}$ in $\mathbb R^2$ , then the graph is closed in $\mathbb R^2$?

If $f:\mathbb R \to \mathbb R$ is an additive function i.e. $f(x+y)=f(x)+f(y) ,\forall x,y \in \mathbb R $ such that $G(f):\{(x,f(x)) : x\in \mathbb R\}$ is a countable intersection of open sets , ...
6
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2answers
176 views

Let $(M,d)$ be a compact metric space and $f:M \to M$ such that $d(f(x),f(y)) \ge d(x,y) , \forall x,y \in M$ , then $f$ is isometry?

Let $(M,d)$ be a compact metric space and $f:M \to M$ such that $d(f(x),f(y)) \ge d(x,y) , \forall x,y \in M$ ; then how to prove that $d(f(x),f(y))=d(x,y) , \forall x,y \in M$ i.e. that $f$ is an ...
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0answers
20 views

How to show $\mathbb{R^2}$ is sequentially connected without path-connectedness

Definitions: Connected: Not separated Separated: If $C$ is a subset of a metric space, then $(A, B)$ is a separation of $C$ if $C = A \cup B$, $A \neq \varnothing$, $B \neq \varnothing$, and ...
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1answer
33 views

If X is complete then $\bar A$ is compact iff $A$ is totally bounded.

Let $A$ be a subset of a metric space $(X,d)$. If X is complete then $\bar A$ is compact iff $A$ is totally bounded. I have done the part that $A$ is totally bounded implies $\bar A$ is compact. But ...
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1answer
35 views

Triangle inequality on the projective space

Given a unit $n$-sphere $\mathbb{S}^n = \{x \in \mathbb{R}^{n+1} : \langle x,x \rangle = 1\}$, we define the set $\mathbb{P}^n = \{[x] : x \in \mathbb{S}^n\}$, where $[x] = \{-x, x\}$, and a function ...
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48 views

Is the following subset of $\mathbb{R^2}$ complete?

I am having troubles with figuring if this space (with the Euclidean metric) is complete or not. $$ \left\{{(x,y)\in \mathbb{R^2}} : x > 0, y \geq \frac{1}{x}\right\}$$ I tried Cauchy sequences, ...
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0answers
18 views

If there is an $\epsilon_x >0$ such that $cl(B(x,ϵ_x))$ is compact for each $x \in X$, then $X$ is complete. [duplicate]

If there is an $\epsilon_x >0$ such that $cl(B(x,ϵ_x))$ is compact for each $x \in X$, then $X$ is complete. We first take a Cauchy sequence $(x_n)$ in $X$ and since it is bounded we get an $x$ ...
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0answers
23 views

Trying to prove Tietze extension theorem

I am trying to prove Tietze extension theorem for metric spaces that is " If $X$ is a metric space , $F$ is a closed set in $X$ and $f:F \to [0,1]$ is a continuous function , then there is a ...
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15 views

property of a separable metric space

I think it is a rather easy question but I don't manage to prove it. \ If $X$ is a separable metric space, then their exists a dense set $x_m, m \in \mathbb{N}$ in X. What I need to prove is the ...
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1answer
17 views

Looking for a counterexample of the gluing lemma for countably infinite collection of closed sets with both domain and range are usual Euclidean space

I was proving the statement of the Gluing lemma that if $X$ is a topological space and $\{A_i:i=1(1)n\}$ is a finite collection of closed sets such that $\cup_{i=1}^n A_i = X$ and if $Y$ is another ...
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22 views

Proving a property for a metric space

Let $(X,d)$ be a metric space. And it also has the property $d(x_1+x_2,y_1+y_2)\leq d(x_1,y_1)+d(x_2,y_2).$ Is it also true that $d(x_1+x_2+...+x_n,y_1+y_2+...+y_n)\leq ...
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1answer
32 views

Matric spaces and σ-closure-preserving bases (Nagata's metrization theorem)

The first two line of the proof say that if $X$ is metrizable (so paracompact) then clearly there is a base $\mathcal{G}= \bigcup_{i\in \mathbb{N}} G_i$ having this property. Paracompactness just ...
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2answers
111 views

$T$ is a linear operator

Define $T: l^2 \mapsto l^2$ by $(Tx)_i = \frac{x_i}i \; \forall i=1,\ldots,n\ldots$. Prove that $T$ is a linear operator with $\|T\|=1$.
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1answer
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Metric spaces and the induced metric

I'm having trouble with the following problem: Suppose $A \subset M$ is a subset of a metric space $(M,d)$. Prove that $U \subset A$ is an open set in the metric space $(A, d_{\vert A \times A})$ if ...
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1answer
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To show that $(X,d)$ is complete. [closed]

Let $(X,d)$ be a metric space such that for each $x \in X$ there exists an $\epsilon _x >0$ with $cl(B(x,\epsilon _x))$ compact, where $cl(A)$ is closure of $A$. To show that $(X,d)$ is complete. ...
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2answers
40 views

The closure of A is closed in X

I am studying for my point-set topology test and want to see if I did this proof right. We are able to assume basic properties of closure... A$\subset$X and (X,d) a metric space Prove that $\bar{A}$ ...
2
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1answer
32 views

Weakly quasisymmetric maps of a connected doubling space are quasisymmetric

I'm currently reading through a few chapters of Juha Heinonen's Lectures on Analysis on Metric Spaces, and I'm having some trouble understanding the finer points of a particular proof. The result is ...
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16 views

Prove if $\{(x,y) \in \mathbb{R^2} : 0\leq x\leq 1, 0\leq y\leq 1\}$ is separated, then there is a separation of $0 \leq z \leq 1$

Definition of separated: Separated: If $C$ is a subset of a metric space, then $(A, B)$ is a separation of $C$ if $C = A \cup B$, $A \neq \varnothing$, $B \neq \varnothing$, and we cannot have ...
5
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1answer
36 views

Subspace of metrizable and separable space is separable

I need to show (using the fact that for metrizable space: space is separable $\iff$ it has got a countable base) that if $X$ is metrizable and separable, then every subspace $Y \subset X $ is ...
0
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1answer
35 views

To show that for some $\epsilon > 0$, $\{x \in X : d(x,A) < \epsilon \ \} \subset U$.

Let $A$ be compact and $U$ be open in a metric space $(X,d)$ such that $A \subset U$. To show that for some $\epsilon > 0$, $\{x \in X : d(x,A) < \epsilon \ \} \subset U$. Let us take a set of ...
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1answer
19 views

Trying to prove $Z(I(A))=\bar A$

For every ideal $I$ of $C[0,1]$ , define $Z(I):=\{x \in [0,1] :f(x)=0 , \forall f \in I\}$ and for every $A \subseteq [0,1]$ , let $I(A):=\{f \in C[0,1] : f(x)=0 , \forall x \in A\}$ . Then ...
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1answer
93 views

Analysis of the ideals of $C[0,1]$

For every ideal $I$ of $C[0,1]$ , define $Z(I):=\{x \in [0,1] :f(x)=0 , \forall f \in I\}$ and for every $A \subseteq [0,1]$ , let $I(A):=\{f \in C[0,1] : f(x)=0 , \forall x \in A\}$ . Then ...
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1answer
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How do I prove that sequence of functions in $C[0, 1]$ defined as below is a Cauchy sequence that does not converge in $C[0,1]$? [closed]

How do I prove that sequence of functions in $C[0, 1]$ defined as below is a Cauchy sequence that does not converge in $C[0,1]$? $$ f_n(t)=\begin{cases} n&t\in\left[0,\frac{1}{n^2}\right]\\ ...
3
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3answers
58 views

Is the set $S=\left\{\left(z_1,z_2\right)\in \mathbb C\times \mathbb C:z_1^2+z_2^2=1\right\}.$ compact?

Consider the set $$S=\left\{\left(z_1,z_2\right)\in \mathbb C\times \mathbb C:z_1^2+z_2^2=1\right\}.$$ Is this set compact in $\mathbb C^2$ ? As $\mathbb C^2$ is a finite dimensional space so a ...
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0answers
34 views

Topologizing ergodic system so that certain function becomes continuous

Let $X$ be a compact metric space and $\mathcal{B}$ its Borel $\sigma$-algebra. Suppose that $(X,\mathcal{B},\mu,T)$ is an invertible ergodic system ($T$ is only a measurable isomorphism, not ...
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2answers
38 views

Is this a separable space?

We have $X = R^n$ and the discrete metric: $d(x,y) = 0$, if $x=y$ and $d(x,y) = 1$ in all other cases. Is this space separable or not? I tried to prove, that the answer for that is no. Let us have ...
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0answers
30 views

Sequence of bounded sequence in metric space

I am reading a paper and bumped at this lemma which I do not know the proof and would like to see some reference. Please suggested me a possible reference. Let $M$ be a metric space and ...
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2answers
53 views

To show that $d(A,B) > 0$ and there exists points $a \in A$ and $b \in B$ such that $d(A,B) = d(a,b)$.

Let $A,B$ be two non-empty, disjoint and compact subsets of a metric space $(X,d)$. To show that $d(A,B) > 0$ and there exists points $a \in A$ and $b \in B$ such that $d(A,B) = d(a,b)$. My Proof: ...
0
votes
1answer
42 views

Prove $\{(x,y) \in \mathbb{R^2} : 0\leq x\leq 1, 0\leq y\leq 1\}$ is connected

Definitions: Connected: Not separated Separated: If $C$ is a subset of a metric space, then $(A, B)$ is a separation of $C$ if $C = A \cup B$, $A \neq \varnothing$, $B \neq \varnothing$, and ...
1
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0answers
41 views

Is the function $f:(a,b) \to \mathbb R$ defined by $f(x):=\dfrac {x-(a+b)/2}{(x-a)(b-x)} , \forall x \in (a,b)$ a homeomorphism?

Is the function $f:(a,b) \to \mathbb R$ defined by $$f(x):=\dfrac {x-\dfrac{a+b}{2}}{(x-a)(b-x)} , \forall x \in (a,b)$$ a homeomorphism ? I have noticed that it is continuous and also noticed that ...
0
votes
1answer
27 views

Metrizable and Metric Topologies

The book I am using for my Introduction of Topology course is Principles of Topology by Fred H. Croom. I am having a hard time clearly understanding the difference between a Metric Topology and a ...
13
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1answer
304 views

Maps from $D^n$ to $D^n$ with a single inverse set are open.

Let $D^n$ denote the closed unit ball in $\Bbb R^n$. In multiple sources proving Brown's generalized Schoenflies theorem (including a version in the original paper), the following consequence of ...
2
votes
2answers
29 views

To show that a function $f : X \to Y$ is continuous iff its graph $G(f) = \{ (x,f(x)) \in X \times Y \}$ is closed in $X \times Y$.

Let $X$ and $Y$ be metric spaces and Y be a compact space. To show that a function $f : X \to Y$ is continuous iff its graph $G(f) = \{ (x,f(x)) \in X \times Y \}$ is closed in $X \times Y$. I have ...
3
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0answers
34 views

To Show that $(\Bbb R,d)$ is bounded and complete but not totally bounded.

On the set $\Bbb R$ of reals consider the metric $d$, given by $d(x,y) = min \{ 1, |x-y| \}$. To Show that $(\Bbb R,d)$ is bounded and complete but not totally bounded. Bounded can be easily verified ...
5
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1answer
105 views

Is $\prod \limits_{i = 1}^{n} [0,1] \subseteq \mathbb R^n$ homeomorphic to the closed unit ball?

Is $\prod \limits_{i = 1}^{n} [0,1] \subseteq \mathbb R^n$ homeomorphic to $\bar B(\theta , 1)$ , the closed ball centered at origin with radius $1$? Can someone please give some reference links to ...
0
votes
1answer
41 views

Is $(\mathbb S^3 \setminus \{0,0,0,1\}) \cap \mathbb R^3 $ homeomorphic with $\mathbb S^2 \times \mathbb R $ ?

Is $(\mathbb S^3 \setminus \{0,0,0,1\}) \cap (\mathbb R^3 \times \{0\})$ homeomorphic with $\mathbb S^2 \times (\mathbb R \times \{0\}\times\{0\})$ ?; here by $\mathbb R^3 \times \{0\}$ I mean ...