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 $f(x)=\tilde{f}(\|x\|)$ and $f$ is continuous, is $\tilde{f}$ continuous?

I am intrigued by this idea that has come to my mind. Let $f:A\subset\mathbb{R}^n\to\mathbb{R}^n$ be a continous funct, either in a point $x_0\in A$ or in all of its domain $A$, whose values only ...
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1answer
19 views

Metric and Absolute value function on $\mathbb R$

I'm contemplating the notion of the absolute value function on $\mathbb R$ as well as of the usual metric on $\mathbb R$. It seems to me that each one of those can be seen in light of the other. ...
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33 views

Give an example of a compact metric space $X$ such that $X$ and $X\times X$ are homeomorphic

Give an example of a compact metric space $X$ such that $X$ and $X\times X$ are homeomorphic. Please suggest me ways on how should I think about this.Its quite sure that $X$ cant be finite. I tried ...
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31 views

Gromov compactness theorem

Reference: this book, page 493. For a compact metric space $X$ define $\text{Cov}(X,\epsilon)= \min \{n \, : \, X \text{ is covered by $n$ closed } \epsilon\text{-balls} \}$ and ...
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1answer
46 views

Proving that a singleton set is both open and closed inside this metric space

Let $(E,d)$ be a metric space and let $a \in E$. Let $\delta(x,y)=\begin{cases} d(a,x)+d(a,y) & x \neq y \\ 0 & x = y \end{cases}$. It can be proved that $\delta$ is a metric on $E$ (I did ...
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52 views

Unions and Intersections of Open Sets are Open

Let $(X,d)$ be a metric space. Prove: the union of any open sets in $X$ is open in $X$ the intersection of a finite number of open sets in $X$ is open in $X$ I could prove the first one but how ...
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29 views

show inverse of $(T^*) = (T^{-1})^*$. [on hold]

Let $H$ be a Hilbert space and $T: H \to H$ a bijective bounded linear operator whose inverse is bounded. Show that inverse of $T^*$ exists and show inverse of $T^* = (T^{-1})^*$.
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7 views

Prove that dual space of projection is equal to normal projection [Hilbert spaces] [on hold]

Let $M$ \subseteq H be a subspace of a Hilbert space H. Prove dual space $(Proj_M)* = ($Proj_M$)
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1answer
43 views

To prove Heine-Borel theorem for $\mathbb R^n$ with usual Euclidean topology

To prove that any closed and bounded subset of $\mathbb R^n$ is compact , I proceed as : Since $\mathbb R^n$ is complete so any closed subset of it is complete . Then I show that any bounded subset of ...
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2answers
41 views

Let $(X,d)$ be an unbounded connected metric space. Let $x \in X$ and $r>0$ be arbitrary then there exists $y \in X$ such that $d(x,y) = r$.

Let $(X,d)$ be an unbounded connected metric space. Let $x \in X$ and $r>0$ be arbitrary then there exists $y \in X$ such that $d(x,y) = r$. We assume on the contrary that there does not exist ...
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28 views

Maximum number of points you can put on grid $ n\times m$ with no equidistant?

Assume we have a grid of $n\times m$ points. and the distance between two rows or two columns is 1 ( unit ). I have a couple of questions related to this grid:- What is the list of possible length ...
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1answer
18 views

Invariance of ball size under translation on the pseudosphere

On the pseudosphere, do the volume of metric balls and area of metric spheres change with the centerpoint? (I'm asking because they don't on the sphere.)
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2answers
37 views

Does every subset of a metric space have an open cover?

I'm having some trouble understanding the concept of compact set (I'm studying from Rudin's Principles of Mathematical Analysis). Does every subset of a metric space have an open cover? Why?
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1answer
70 views

$f(X)$ is uncountable and hence $X$ is uncountable.

My question: let $f : X \to \Bbb R$ be a non-constant continuous function on a connected metric space and assume that $f(X)$ is uncountable; then $X$ is uncountable. We know continuous image of a ...
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0answers
15 views

small expected contraction embedding into trees?

I learned FRT theorem for probabilistic metric embedding into trees: For any finite metric d, there exists a distribution over non contracting, small expected expansion tree metrics. The theorem can ...
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1answer
31 views

covering number and compactness

The following picture is what I extracted from the end of page 7 in http://www-personal.umich.edu/~romanv/papers/non-asymptotic-rmt-plain.pdf My confusion is on the blue part: in 1-dimensional ...
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1answer
21 views

Show that dual space of $R^n$ with norm 3 is equal to the $R^n$ with norm 1.5.

How can one prove that dual space ($R^n$,$||.||_3$)*= ($R^n$,||.||1.5). How to go about using the holder's inequality? Any help will be appreciated! Hint: I know I've to use holder inequality to make ...
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3answers
59 views

What is wrong with my proof that $f^{-1}(S)$ is open?

Let $X$ and $Y$ be metric spaces, $f: X \to Y$ is continuous, $S \subset Y$, and $S$ open. Prove that $f^{-1}(S)$ is open, where $f^{-1}(S) = \{x \in X : f(x) \in S\}$. If $x \in f^{-1}(S)$, then ...
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1answer
42 views

A set $A \subset l_1$ is compact

A set $A \subset l_1$ is compact if and only if $A$ is closed and bounded and given any $\epsilon >0$, there exists $n_0$ such that $\sum_{k=n}^{\infty} |x_k| < \epsilon$ for all $n> n_0$ and ...
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24 views

$\{U_{\alpha} \}_{\alpha \in I}$ collection of connected sets , for every $U_{\alpha}$ , $\exists U_{\beta}\ne U_{\alpha}$ not mutually disjoint

A probable further strengthening of $\{U_{\alpha} \}$ is a collection of connected sets in a metric space such that no two connected sets in the collection is disjoint ... If $\{U_{\alpha} \}_{\alpha ...
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1answer
18 views

Closed set and set, closed in $\mathfrak M$

I've read in my textbook that a set $A$ is called closed if it contains its limit points, i.e. $A'\subseteq A$. But then, coming to next chapter, I came across a term of set $B$, closed in metric ...
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2answers
77 views

Use definitions to show $[0, 1) × [0, 1)$ is neither an open nor closed subset of $\Bbb{R^2}$.

Show, from the definitions of open and closed sets, that when using the standard Euclidean metric, [0, 1) × [0, 1) is neither an open nor closed subset of $\Bbb{R^2}$. From what I understand, a set ...
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3answers
157 views

Proving Any connected subset of R is an Interval

Common Proof: Suppose $S$ is not an interval of $R$. Then by Interval Defined by Betweenness, $∃x,y∈S$ and $z\in R∖S$ such that $x<z<y$. Consider the sets $A_1=S∩(−∞,z)$ and ...
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21 views

Suppose that $f:X \rightarrow X'$ is a one-one correspondence of metric spaces such that $f$ is uniformly continuous and $f^{-1}$ is continuous.

Suppose that $f:X \rightarrow X'$ is a one-one correspondence of metric spaces such that $f$ is uniformly continuous and $f^{-1}$ is continuous. Prove that if $(y_n)$ is a convergent sequence in $X'$ ...
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32 views

$E$ compact, real-valued $f : E \to \mathbb{R}$ continuous iff graph is compact - is real valued necessary?

Problem The graph $G$ of $f$ is defined as the points $(x, f(x))$ for $x \in E$. Suppose $E \subset \mathbb{R}$ is compact, then $f : E \to \mathbb{R}$ is continuous iff its graph is compact. ...
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2answers
32 views

$\{U_{\alpha} \}$ is a collection of connected sets in a metric space such that no two connected sets in the collection is disjoint …

If $\{U_{\alpha} \}$ is a collection of connected sets in a metric space such that no two connected sets in the collection is disjoint , then is the union of all the sets in the collection connected ...
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1answer
34 views

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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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
73 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 ...
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1answer
36 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 ...
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1answer
45 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
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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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1answer
28 views

If $(X,d)$ is a finite metric space , then every ideal of $C(X, \mathbb R)$ is generated by an idempotent ? [on hold]

If $(X,d)$ is a finite metric space , then is it true that every ideal of $C(X, \mathbb R)$ is generated by an idempotent ?
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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
23 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
75 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 ...
0
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2answers
13 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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14 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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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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1answer
17 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
31 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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+50

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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1answer
100 views
+50

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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17 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
32 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 ...
0
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1answer
33 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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3answers
45 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, ...
0
votes
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
22 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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54 views

erwin kreyszig introductory functional analysis with applications page 82 exercise 4 [on hold]

We know that the space $S$ consists of the set of all (bounded or unbounded ) sequences of complex number and the metric $d$ defined by : $$d(x,y)= \sum_{j=1}^{+\infty} \frac{1}{2^j} ...