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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How to write Holder's inequality for random vectors?

For $1 < p,q < \infty$ satisfying the constraint $1/p + 1/q =1$ and for $X, Y$ random variables such that $\mathbb E [\vert X \vert ^p ], \mathbb{E} [\vert X \vert ^q ] < \infty $ we have ...
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How to show this is a metric?

$d_1(x_1,y_1)$ and $d_2(x_2,y_2)$ are metric on $X$ and $d(x,y)$ is defined as: $$d(x,y)= \sqrt{d_1(x_1,y_1)^2+d_2(x_2,y_2)^2}.$$ I am trying to show this is a metric. Can you give me some clue about ...
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Prove that if a subset $A$ of a metric space is bounded then the closure of $A$ is bounded and diam(A) is equal to the diam(cl(A)).

Prove that if a subset $A$ of a metric space is bounded then the closure of $A$ is bounded and the diameter of $A$ is equal to the diameter of the closure of $A$. This is the question I am working on ...
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63 views

$X$ is Frechet Compact iff $X$ is compact.

I have done the proof that $1)\ X$ is Frechet Compact iff $X$ is sequentially compact. $2) \ X$ is sequentially compact iff $X$ is compact. Thus we can conclude that $X$ is Frechet Compact iff ...
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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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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.($|X|>1)$ Please suggest me ways on how should I think about this.Its quite sure that $X$ cant be ...
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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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379 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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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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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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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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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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36 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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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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$f(X)$ 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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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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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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205 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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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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64 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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$\{U_{\alpha} \}_{\alpha \in I}$ 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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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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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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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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$E$ 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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$\{U_{\alpha} \}$ 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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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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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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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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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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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 ...
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60 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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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 $'s are real valued ...
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$(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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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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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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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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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 ...
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1answer
46 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 ...