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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253 views

Show that $\displaystyle\prod_{\Bbb{N}} \Bbb{R}$ with the box topology is Hausdorff but not metrizable.

Show that $\displaystyle\prod_{\Bbb{N}} \Bbb{R}$ with the box topology is Hausdorff but not metrizable. $\Bbb{R}$ must be Hausdorff. For $x_1, x_2 \in \Bbb{R}$ (where $x_1 \not= x_2$), if $d$ ...
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3answers
137 views

Is the set of extended real-valued numbers open or closed

If I assume that my topology is defined on the extended real-valued numbers, then $\mathbb{R}\cup\left\{-\infty,+\infty\right\}=\left[-\infty,+\infty\right]$, acting as my entire space, is both open ...
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1answer
275 views

Proving the existence of a non-monotone continuous function defined on $[0,1]$

Let $(I_n)_{n \in \mathbb N}$ be the sequence of intervals of $[0,1]$ with rational endpoints, and for every $n \in \mathbb N~$ let $E_n=\{f \in C[0,1] : f \:\text{is monotone in}\: I_n\}$. Prove that ...
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4answers
66 views

Prove B is a closed subset of X given the f and g are continous?

Let $(X;\rho)$, $(Y;\sigma)$ be metric spaces. Let $f,g : X \to Y$ be continuous. Prove that the set $B=\{x\in X: f(x)=g(x)\}$ is a closed subset of $X$
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1answer
64 views

Connected components of a given space

For every $n \in \mathbb N$, let $A_n=\{\frac{1}{n}\}\times[0,1]$, and let $X=\bigcup_{n \in \mathbb N} A_n \cup \{(0,0),(0,1)\}$. Prove that: i)$\{(0,0)\}$ and $\{(0,1)\}$ are connected components of ...
2
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1answer
384 views

looking for proof that this uniformly bounded sequence of functions has no pointwise convergent subsequence

Math people: I couldn't find a similar question, so here goes: I would like to prove the fact (?) that the sequence of functions $(f_n) \subset C([0,1])$ defined by $f_n(x)=\sin(nx)$ does not have a ...
2
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2answers
213 views

Prove that the identity map $(C[0,1],d_1) \rightarrow (C[0,1],d_\infty)$ is not continuous

$$d_\infty = \max|x_i - y_i|$$ $$d_1 = \sum_{i=1}^n |x_i - y_i|$$ The first part of this question was to prove that the identity map $$(C[0,1],d_\infty) \rightarrow (C[0,1],d_1)$$ is continuous, ...
0
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1answer
57 views

Real Analysis Closed and Bounded Set Question

Suppose K is a nonempty closed and bounded subset of a metric space X and x $\in$ X. Show the following hypothesis fails: There is a p $\in$ K such that, for all other q $\in$ K, d(p,x) $\leq$ d(q,x). ...
2
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2answers
506 views

Show that the topological space ( X, $\tau$ ) is not metrizable

For the topological space ( X, $\tau$ ), with X = {0, 1} and $\tau$ = { $\emptyset$ , {0}, {0,1} } , prove that ( X, $\tau$ ) is not metrizable. I know intuitively it can't be but don't know how to ...
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2answers
46 views

convexity and the interior sphere condition

Consider $\Omega $ a open, convex bounded subset of $R^n$. Let $x_0 \in \partial \Omega$. I believe that exists a open ball $B \subset \Omega$ such that $\partial B \cap \partial \Omega = \{ x_0 \}$. ...
4
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1answer
67 views

Free Metric Space?

Do free metric spaces exist? Ie.: An object in the category of metric spaces and lipschitzian maps. If so would these be the complete metric spaces, since they satisfy a similar universal property? ...
1
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1answer
42 views

Whether a function$d(m,n)=\left\vert\frac{1}{m}-\frac{1}{n}\right\vert$ metrics

I saw in a magazine the following example" Whether a function $d(m,n)=\left\vert\frac{1}{m}-\frac{1}{n}\right\vert,$ where $m,n\in\mathbb{N}$ metrics. I know that map $d:XxX\rightarrow\mathbb{R}$ ...
0
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1answer
425 views

Show that $C[a,b]$ is a complete space under the metric $d(f,g)=\sup_{t\in [a,b]}|f(t)-g(t)|$.

$C[a,b]$ is a normed vector space of all continuous complex valued functions on $[a,b]$, with supremum norm $$\|f\|_\infty=\sup_{t\in [a,b]}|f(t)|.$$ The metric induced by the norm is ...
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2answers
109 views

Open sets of sequences

Let $M$ denote the space of sequences $(x_n)$ where $x_n \in\{0,1\}$ for each $n$. Let $$d\colon M\times M\rightarrow\mathbb{R}\colon ((x_n),(y_n))\mapsto\sum_{i=1}^{\infty}|x_i-y_i|2^{-i}$$ be the ...
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1answer
744 views

Complete metric spaces - continuous functions

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces with $(Y,d_Y)$ bounded. Let $C(X,Y)$ denote the set of all continuous functions from $X$ to $Y$. Let $d$ be the uniform metric on $C(X,Y)$, i.e. $d(f,g) = ...
5
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3answers
159 views

Extending a connected open set

Assume $\emptyset\neq V\subseteq U\subseteq\mathbb{R}^n$ are open and connected sets so that $U\setminus\overline{V}$ is connected as well. Given any point $x\in U$, is there always a connected open ...
0
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2answers
89 views

the power series converges in compact convergence topology

Consider the sequence of functions $f_{n}: (-1,1) \rightarrow R$ defined by:$$f_{n}(x) = \sum_{k=1}^{n}{kx^{k}}$$ a) Prove that $(f_{n})$ converges in the topology of compact convergence, ...
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1answer
44 views

$B(R,R)$ is not closed in the topology of compact convergence

I'm doing this exercise in Munkres book, and got no clue to solve this problem. Help someone can help me. Let $B(R,R)$ be the set of bounded functions $f: R \rightarrow R$. Prove that ...
0
votes
1answer
359 views

relative compact implies totally bounded?

Let $M$ be a metric space. It's always true that if $A$ is relative compact (i.e $\bar{A}$ is compact) then $A$ it's also totally bounded?. I tried to proved it, considering the finite subcovering of ...
2
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1answer
71 views

Some questions on convex sets.

Are all bounded closed convex sets in a metric space $(M,d)$ compact? or if not are they complete? The positive definite matrices form a convex set (Why does a positive definite matrix defines a ...
3
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1answer
97 views

Metric spaces and limit points question?

Let $X, d$ be a metric space. For each $x \in X$ and nonvoid $A, B \in X$, define $$d(x, A) = \inf\{d(x, a) : a \in A\}$$ and $$d(A, B) = \inf\{d(a, b) : a \in A, b \in B\}$$ Prove that $d(x, A) = 0$ ...
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1answer
64 views

Metric on an infinite dimensional space with equivalence relation.

In analyzing a problem I've come across a space defined by the following equivalence relation: $(\cdots, x_{-2}, x_{-1}, x_0, x_1, x_2, \cdots) \sim (\cdots, z^{-2}x_{-2}, z^{-1}x_{-1}, x_0, zx_1, ...
4
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0answers
194 views

Completeness of a metric space with the Hausdorff metric

Let $(Y,d)$ be a metric space and let $K(Y)$ denote the set of all non-empty compact subsets of $Y$. This collection is a metric space when equipped with the Hausdorff distance $h$. I want to prove ...
3
votes
1answer
75 views

What type of convex constraint is defined by SQRT?

Let $A$ be an $n \times n$ positive semidefinite matrix and $\forall k, x_k \in \mathbb{R}^n$. The distance with respect to this matrix is defined as $ \|x_i -x_j\|_A := ...
0
votes
2answers
203 views

Prove that open subspace of a topologically complete space is topologically complete

I'm trying to prove that an open subspace of a topologically complete space is topologically complete. I follow the hint in the book. We defined $\phi : U \rightarrow R$ by the equation $$\phi(x) ...
11
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2answers
687 views

A metric space such that all closed balls are compact is complete.

I am trying to solve the following exercise: Let $(X,d)$ be a metric space that has the property that for any $x\in X$ and $r>0$, the closed ball $$\bar{B}(x,r):=\{y\in X:d(x,y)\leq r\}$$ ...
0
votes
1answer
40 views

$\sup_n \inf_{y \in X} (F(y) + n d(x,y))= F(x)$

Let $(X,d)$ be a metric space and $F : X \rightarrow [0, +\infty)$ a lower semicontinuous function. Then $$ \sup_n \inf_{y \in X} (F(y) + n d(x,y))= F(x). $$ Is this true? Intuitively it works since ...
5
votes
3answers
168 views

Does $d(x_{n+1},x_n)<d(x_n,x_{n-1})$ imply anything?

Let $(X,d)$ be a complete metric space, $(x_n)_{n\in\mathbb{N}}\subset X$ such that $d(x_{n+1},x_n)<d(x_n,x_{n-1})$ for all $n\in\mathbb{N}$. Since I cannot construct such sequence which is not ...
2
votes
1answer
579 views

Continuous images of Cauchy sequences are not necessarily Cauchy

Could you please provide an example for two metric spaces $X,Y$, a continuous function $f$ that maps $X$ to $Y$ and a Cauchy sequence in $X$, which is not mapped to a Cauchy sequence in $Y$ by $f$? ...
6
votes
2answers
546 views

Every open ball is connected

Let $(X,d)$ be a metric space such that for all $x \in X$ and all $r>0$, $\overline{B(x,r)} = \{y \in X \mid d(x,y)\leqslant r\}$ Show that every open ball of $X$ is connected. Note- I was trying ...
1
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2answers
53 views

Showing that a set is open

Endow $R^2$ with the metric $d(a,b)$ ={ $max{|a_1-b_1|,|a_2-b_2|}$} where $a$=$(a_1,a_2)$ and $b$=$(b_1,b_2)$. Show that $S$={${a \in R^2|a_1^2+a_2^2<1}$} is open in $R^2$ with this metric. $S$ ...
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votes
1answer
62 views

Showing that a set is open/closed

$\def\R{\mathbb R}$ Is the set $$S=\{(x_1,x_2,x_3) \in \R^3 \mid e^{x_1} + x_2^2 <x_3 \} \subset \R^3$$ open or closed? My attempt: Let $f:\R^3 \to \R$, $f(x_1,x_2,x_3)$ =$e^x_1 + ...
0
votes
1answer
45 views

Showing that a set is closed

Show that the set $S=\{a \in \mathbb{R}^3\,| \,a_1 +a_3^2 \sin(a_1+a_2)\geqslant a_3\}$ in closed in $\mathbb{R}^3$ with the euclidean metric. I know that I would probably have to show that the ...
0
votes
1answer
114 views

Infinite intersection and limits

I'm having difficulty understanding the relationship between a limit and an infinite intersection. Any help would be greatly appreciated. Specifically, suppose we take any non-increasing sequence of ...
0
votes
1answer
34 views

Extract a converging geodesic from a sequence

Let $(X, d)$ be a compact, complete, separable metric space, and $g_n$ a sequence of constant speed geodesic with the same endpoints, i.e. continuous maps $g_n : [0,1] \rightarrow X$ such that $$ ...
1
vote
3answers
105 views

Continuity of metric [duplicate]

I recently came across this definition: Let $(X,d)$ be a metric space and $A$ be a nonempty subset of $X$. For each $x\in X$ we define a distance from $x$ to $A$ by the equation $d(x,A)=\inf\{d(x,a) ...
0
votes
1answer
44 views

Existence of a geodesic in a complete separable metric space

If I have $X$ a complete separable metric space, $x, y \in X$ arbitrary points, how can I define a constant speed geodesic, i.e. a continuous map $g : [0,1] \rightarrow X$ such that $$ d(g(t), g(s)) = ...
0
votes
1answer
34 views

Compact space with $x_{n_j} \to x $ for all conv. subsequences

Given a compact metric space $(X,d)$ with sequence $(x_n)_n \subseteq X$ and every convergent subsequence of $(x_n)_n$ converges against $x$. How can I show that $x_n \to x$? Hints are welcome! ...
1
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1answer
158 views

Show that $f:[0,1] \to [0,1]$ is continuous if $f(x) = x^{1/k}$ for any $k \in \mathbb N$

I'm very confused right now and I want to apply the theorem that says " A mapping f of a metric space $X$ into a metric space $Y$ is continuous on $X$ if and only if $f^{-1}(V)$ is open in $X$ for ...
1
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1answer
220 views

Showing that $(S,d)$ is a metric space for $S = (0,1]$ and $d(x,y) = |1/x - 1/y|$

Let $S$ be a half-open interval $(0,1]$. If we define $d$ on $S$ by $$d(x,y) := \left|{1\over x} - {1\over y}\right|\;,$$ then show that $d$ is a metric on $S$. Also, prove that $(S,d)$ is a ...
1
vote
1answer
211 views

Show that the following is a metric on $\mathbb R$

Is the following just a matter of showing the 3 properties that make up a metric?? Define d on $\Bbb R\times\Bbb R$ by $d(x,y)=\min \{1,|x-y|\}$. Show that $d$ is a metric on $\Bbb R$ $d(x,y)=0$ ...
1
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2answers
79 views

how to prove this is a metric given the following conditions

I need help wrapping my head around the concepts of metrics and how to prove that something is a metric. For example, prove that if $p_1$ and $p_2$ are metrics in $X$, then $p_1 + p_2$ and $\max\{p_1, ...
0
votes
1answer
66 views

How to find the closure of a subset

How do I start find the closure of a subset? Let's say I'm given a list, such as $$A=\left\{\frac12,\frac13,\frac23,\frac14,\frac24,\frac34,\frac15,\frac25,\frac35,\frac45,\cdots\right\}$$ using the ...
1
vote
1answer
171 views

Homeomorphism of closed intervals

One can prove that if $f: [a,b] \to [f(a), f(b)]$ is continuous and monotone increasing that then it is a homeomorphism. The only part one might have to think about at all is that $f$ is open but that ...
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vote
2answers
514 views

Sequence has a convergent subsequence in R^n

Suppose A is a closed and bounded subset of R^n. Let {ak} be a sequence in A. Thus, the elements of {ak} are: (a11,a12,...,a1n), (a21,a22,...,a2n), ... ... (ak1,ak2,...,akn), ... We are not sure if ...
4
votes
1answer
547 views

Existence of limit for convergence by measure for Cauchy-in-measure sequence+completeness of metric space?

Sorry if this is the wrong place to put it. But this question come from a graduate level textbook and seems pretty hard to me, so I hope this is a good place. Anyway, this come from the book Real ...
2
votes
0answers
46 views

Finding an isometry given the distance between points

If $A,B,C,D \in \mathbb{R}^n$, such that $d(A,B)=d(C,D)$, then exists an isometry $f:\mathbb{R}^n\rightarrow \mathbb{R}^n $, such that $f(A)=C, f(B)=D$. Thanks a lot for the help.
0
votes
1answer
396 views

Proving subset is not connected iff there exist open sets in X

Prove that E $\subseteq$ X is not connected if and only if there exist open sets $A, B \subseteq X$ such that $E \subseteq A ∪ B, A ∩ B$ = $\emptyset$ and $E ∩ A$ and $E ∩ B$ are both nonempty. $X$ ...
0
votes
1answer
96 views

Flat space Minkowski metric

I am having some problem understanding the why in Minkowski spacetime, the continuity equation is written as $$\partial_\mu J^\mu=0.....................(*)$$ Physically, I know that $$\partial_t ...
0
votes
2answers
68 views

Proving sequential compactness from open cover compactness.

Let $(\mathcal M,d)$ be a metric space and $A\subset\mathcal M$. The following types of compactness are equivalent: (i) Each open cover of $A$ contains a finite subcover. (ii) $A$ is sequentially ...