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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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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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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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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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 $\mathbb{R^...
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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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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 continuous ...
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71 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 \...
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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 , ...
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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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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
53 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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55 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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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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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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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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26 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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72 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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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 d(x_1,y_1)+d(x_2,y_2)+...+...
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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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$T$ 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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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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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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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 $\bar{A}$ ...
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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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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 ...
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45 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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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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106 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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How to prove continuity in Baire space?

Let $X=(\omega^\omega,d)$ be Baire space with the metric $d$ defined in assignment $1$. Define a function $G:X\to X$ by letting, for $f\in X$, the function $G(f)$ be defined by: $$(G(f))(n)=\begin{...
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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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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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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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38 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 $K^{(n)}={x^...
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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: ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
51 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 $\{(...
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1answer
34 views

Can we prove, without invoking invariance of domain, that $\mathbb R$ and $\mathbb R^2$ are not homeomorphic?

Can we prove, without invoking invariance of domain, that $\mathbb R$ and $\mathbb R^2$ are not homeomorphic, or equivalently, that no open set of $\mathbb R$ is homeomorphic to an open set of $\...
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1answer
78 views

Existence of continuous bijective function $f:[0,1] \times [0,1] \to [0,1] $ ? Continuous and only injective and continuous and olny surjective?

Does there exist any continuous bijective function $f:[0,1] \times [0,1] \to [0,1] $ , where $[0,1]$ is equipped with usual Euclidean metric of $\mathbb R$ and $[0,1] \times [0,1]$ is equipped with ...
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2answers
166 views

Discrete subset of $\mathbb R^2$ such that $\mathbb R^2\setminus S$ is path connected.

Let, $S\subset \mathbb R^2$ be defined by $$S=\left\{\left(m+\frac{1}{2^{|p|}},n+\frac{1}{2^{|q|}}\right):m,n,p,q\in \mathbb Z\right\}.$$ Then, which are correct? (A) $S$ is a discrete set. (B) $\...
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28 views

How to prove d is metric if prime number is involved?

Please have a look at this question. Can anyone please tell me how to approach when dp(x,y) /= 0? Does this mean if x= y, then d=0, otherwise d = p^-k? or only if p^k satisfy the following condition? ...
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30 views

Periodic Points of $h:=f\times g: [0,1]^2\to[0,1]^2$ for Continuous $f,g$

Question: Let $f,g:[0,1]\to[0,1]$ be continuous, $h:=f\times g:[0,1]^2\to[0,1]^2,$ $(a,b)\mapsto(f(a),g(b))$. Then "Period Three Implies Chaos" applies to $h$, while Sharkovskii's Theorem does not. ...
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1answer
37 views

To show that for each $\epsilon > 0$, there exists an infinite set $B \subset A$ such that diam$(B) < \epsilon$.

If $A$ is an infinite subset of a totally bounded metric space $(X, d)$, then to show that for each $\epsilon > 0$, there exists an infinite set $B \subset A$ such that diam$(B) < \epsilon$. ...
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1answer
65 views

Counterexamples for the Converse of “Topological Conjugacy Implies Equal Topological Entropy”

Question: I would like to find two topological dynamical systems that are not topologically conjugate but nevertheless have the same topological entropy. Two topological dynamical systems $f:X\to X,g:...
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1answer
23 views

$U$ be open in $X$ and $A:=X \setminus U$ , is the function $f: U \to \mathbb R ; f(x):=dist (x,A) , \forall x \in U$ injective ?

Let $(X,d)$ be a metric space , $U$ be open in $X$ and $A:=X \setminus U$ , is the function $f: U \to \mathbb R$ defined by $f(x):=dist (x,A) , \forall x \in U$ injective ? If not , then do we need ...