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.

learn more… | top users | synonyms (1)

0
votes
2answers
19 views

Sequence of partial sums of e in Q is a Cauchy sequence.

Verify that $X_n= \{ \sum_{i=0}^n$ $\frac{1}{i!}$} is a Cauchy sequence in $Q$ with the Euclidean metric. I can't figure out how to find an $N$ that makes this work. I figure that $d(x_n,x_m) < ...
0
votes
1answer
40 views

Establish if $g_n (\alpha)=\int_a^b \ \alpha(x) \ \sin (nx) \ \cos(nx) $ converges uniformly

$$X=\{ \alpha:[a,b] \rightarrow \mathbb{R} \}$$ $\alpha''$ exists and it is continuous $$\exists \ K>0 \ : \forall \ x \in [a,b], \forall \alpha \in X: \\ \ \\ \rvert \alpha(x) \rvert, \rvert ...
0
votes
0answers
28 views

Natural embedding of Q with the Euclidean metric in R with the Euclidean metric is an isometric embedding.

The book I'm reading states this: The natural embedding of $Q$ with the Euclidean metric in $R$ with the Euclidean metric is an isometric embedding. What is the "natural embedding" of Q with the ...
0
votes
1answer
17 views

Banach fixed point theorem for a function $f_k(x) = k(x+1/x)$

Suppose $X = [1,\infty)$. The function $f_k(x) = k(x+\frac{1}{x})$ where $k\in(0,1)$ is a contraction on $X$, furthermore, $X$ is complete and $f:X\rightarrow X$. So all the requirements for the ...
3
votes
1answer
18 views

If for every $\epsilon>0$ there exist infinitely many $n$ such that $d(x_n,c)<\epsilon$ then $(x_n)$ has a subsequence converging to $c$.

Suppose $(X,d)$ is a metric space. I am trying to show that: If for every $\epsilon>0$ there exist infinitely many $n$ such that $d(x_n,c)<\epsilon$ then $(x_n)$ has a subsequence converging to ...
5
votes
1answer
104 views

Borel measurability of a subset of a product space

Let $X$ and $Y$ be compact metric spaces and let $\mathcal B_X$ and $\mathcal B_Y$ be their respective Borel $\sigma$-algebras. Let $\mu$ be a Borel probability measure on $X$ and let $\mathcal ...
0
votes
2answers
21 views

Pointwise convergence of Lipschitz functions from a compact space implies uniform convergence

Let $(f_n)$ be a sequence of $1$-Lipschitz functions from $(X, d_X)$ to $(Y,d_Y)$ where the first one is compact and the latter is complete (I am not sure if this matters). Let $f_n \to f$ pointwise. ...
6
votes
2answers
99 views

$(f(x_n))_{n\in \mathbb{N}}$ converges for all continuous $f$. Does $(x_n)_{n \in\mathbb{N}}$ converge?

Let $(S, d)$ be a metric space and $(x_n)_{n\in \mathbb{N}}$ a sequence in $S$. If $(f(x_n))_{n\in \mathbb{N}}$ converges for all continuous $f:S\to\mathbb{R},$ does it follow that $(x_n)_{n\in ...
1
vote
1answer
35 views

Does there exist a metric space of cardinality aleph-two that has a countable epsilon cover?

Does there exist a metric space $(M,d)$ such that $|M|=\aleph_2$ but there exists a countable $\epsilon$-cover of $M$?
0
votes
1answer
26 views

Cauchy sequences are bounded

As $\{x_n\}$ is a Cauchy sequence, there exists a positive integer $N$, such that for any $n \geq N$ and $m \geq N$, $d(x_n,x_m) \lt 1$; that is, $|x_n-x_m| \lt 1$. Put $M = |x_1| + |x_2| + |x_3| + ...
0
votes
1answer
47 views

Conformal map is an isometry

I have the upper half-plane $\mathbb H$ with the metric given by $$\mathrm ds^2=\frac{1}{y^2} (\mathrm dx^2+\mathrm dy^2)$$ and the unit disk $\mathbb D$ with the metric given by $$\mathrm ...
1
vote
3answers
60 views

Does there exist a continuous function $g:S^1 \to S^1$ such that $(g(z))^2=z , \forall z \in S^1$?

Let $S^1:=\{z \in \mathbb C:|z|=1\}$ ; does there exist a continuous function $g:S^1 \to S^1$ such that $(g(z))^2=z , \forall z \in S^1$ ?
1
vote
2answers
23 views

Compactness of infinite union under these conditions

Assume I have an infinite sequence $(S_k)_{k\in\mathbb N}$ of sets $S_k\subset \mathbb R^n$, assume that all the $S_k$ are compact with respect to the topology induced by some metric $d:\mathbb ...
1
vote
2answers
44 views

Is there anything special about the below finite metric space? See below for details.

I am a high school student who has been playing around with certain mathematical ideas, most recently metric spaces, and I believe I have just "defined" if you will, the following metric space: Metric ...
1
vote
1answer
18 views

What values are used for the counter $k$ in this proof involving a convergent subsequence?

Lemma Suppose that $(x_n)$ is a Cauchy sequence in a metric space $(X,d)$, and $x\in X$. Also suppose $(x_{n_k})$ is a subsequence of $(x_n)$ such that $x_{n_k}\to x$ as $k\to \infty$. Then $x_n\to ...
4
votes
1answer
40 views

Why do metric spaces that produce the same topology have different number theoretical difficulties?

Consider finding a a point with rational distance to the corners of unit square. Under the Euclidean metric this is very hard. (unsolved) Under the "city block" or taxicab metric this is very easy ...
4
votes
2answers
48 views

Does there exist a compact metric space $X$ containing countably infinitely many clopen subsets?

From this Clopen subsets of a compact metric space we know that any compact metric space $X$ contains at most countably many clopen subsets ; my question is : Does there exist a compact metric space ...
0
votes
1answer
36 views

Show that $\bigcap_{r>0}B(a,r)=\bigcap_{n=1}^\infty B(a,\frac{1}{n})=\{a\}$

Let $(M,d)$ be a metric space. Show that $\bigcap_{r>0}B(a,r)=\bigcap_{n=1}^\infty B(a,\frac{1}{n})=\{a\}$ where $B(a,r)$ is a ball with center in $a$ and radius $r$. My attempt: Set $0<r\leq ...
1
vote
1answer
28 views

Co-ordinate transformation of metric

In a past exam paper that I am using to prepare for my upcoming finals, I have encountered the following question (paraphrased): Given the metric: $$\mathrm{d}s^{2} = ...
7
votes
1answer
70 views

$X,Y$ be metric spaces , $f:X \to Y$ be a continuous and closed map , then the boundary of $f^{-1}(\{y\})$ is compact for every $y \in Y$ ?

Let $X,Y$ be metric spaces , $f:X \to Y$ be a continuous and closed map , then is it true that the boundary of $f^{-1}(\{y\})$ is compact for every $y \in Y$ ?
1
vote
3answers
45 views

Homeomorphism from $(0,1)$ to $\mathbb{R}$

I want to show that $(0,1)$ is homeomorphic to $\mathbb{R}$ by finding a homeomorphism between the two. I think the function will be related to $tan(x)$ but I'm stuck on how to modify it to fit the ...
1
vote
2answers
35 views

Subset of separable metric space can have at most a countable amount of isolated points

Let $(X,d)$ be a separable metric space. Prove that every subset $Y \subset X$ can have at most a countable amount of isolated points. Attempt at proof: Let $Y$ be an arbitrary (non-empty) subset of ...
0
votes
2answers
35 views

Topological finer and separability

I have this question: Let $(X, d_1)$ and $(X,d_2)$ be two metric spaces. Suppose $d_1$ is topologically finer than $d_2$. What is the relationship between these two statements? (i) $(X,d_1)$ is ...
0
votes
0answers
15 views

Differentiable version of Urysohn's lemma

Let $A,B$ be disjoint non-empty closed sets in $\mathbb R$ , then does there exist a differentiable function $f:\mathbb R \to [0,1]$ such that $f(A)=\{0\} , f(B)=\{1\}$ ? If the answer to the previous ...
0
votes
0answers
19 views

Given $(X,d)$ is a metric space , then the following statements are equivalent.

We need to show that if $(X,d)$ is disconnected => there exists two non-empty disjoint subsets $A$ and $B$ both open in X s.t $X= A \cup B$. I was able to prove the disjoint part , now we need to ...
0
votes
1answer
10 views

Name for function that is Lipschitz continuous over partitioning of input space

Let $f: X \to \mathbb R$ and $(X,d)$ be a metric space. Let $P=\{P_1,P_2,\dotsc\}$ be a countable partitioning of $X$. I would like to assume that $f$ is Lipschitz continuous on $(P_i,d)$ for all $P_i ...
0
votes
0answers
39 views

Does every metric space have a countable $\epsilon$-cover? [closed]

Does there exist a metric space that does not have a countable $\epsilon$-cover?
0
votes
0answers
21 views

$X \subseteq M(n,\mathbb C)$ and $S:=\{x \in \mathbb C : x$ is an eigen value of some matrix in $X\}$ ; does $X$ closed implies/if $S$ is closed?

Let $X \subseteq M(n,\mathbb C)$ and $S:=\{x \in \mathbb C : x$ is an eigen value of some matrix in $X\}$ ; does the closed-ness of any one of $S$ or $X$ implies that the other set is also closed ?
0
votes
1answer
19 views

Open and closed sets in a $\infty$-metric space

Denote by $\mathcal{H}$ the set of continuous maps $ h:\mathbb{R}\rightarrow \mathbb{R}$ such that $h(0)=0$. We endow $\mathcal{H}$ with the supremum metric $$ \widehat{d}(f,g)=\sup\{\vert ...
0
votes
2answers
33 views

Constructing a sequence of functions, not Cauchy

I'm working in the set $B = \{ f \in C[0,1] : \int_0^1 f(x)dx \leq 1\}$. I'm constructing an argument to show that there exists at least one sequence that has a subsequences satisfying the property ...
1
vote
1answer
24 views

$X \subseteq M(n,\mathbb C) ; |X|>1 ; $ connected /path connected , what about $S:=\{x \in \mathbb C : x $ is an eigenvalue of some matrix in $X\}$?

Let $X \subseteq M(n,\mathbb C)$ be a set with more than one element ( I am also considering $M(n,\mathbb R) \subseteq M(n,\mathbb C)$ and $S:=\{x \in \mathbb C : x $ is an eigenvalue of some matrix ...
0
votes
3answers
47 views

If a set is open in one metric it is open in another?

Ive been struggling to grasp a certain situation involving metric spaces and was wondering if anyone could be of any help. In the notes for my module on metric spaces I have the following "If two ...
2
votes
1answer
28 views

A discrete metric space is complete

We can read here that every discrete metric space (where the topology is the same as the discrete topology, i.e. where all the singletons are open) is complete, but an example bothers me because I ...
2
votes
1answer
41 views

Connectedness of the sets $\{A \in M(n,\mathbb R) : \mathrm{rank}(A) =m\}$ and $\{A \in M(n,\mathbb R) : \mathrm{rank}(A)\ge r\}$

Let $r>0$ , I know that $\{A \in M(n,\mathbb R) : \mathrm{rank}(A) < r\}$ is path connected in $M(n,\mathbb R)$ . My question is ; for positive integer $m < n$ , is the set $\{A \in ...
1
vote
1answer
38 views

If $X$ is compact and $T:X \to Y$ continuous and bijective, show that $T$ is homeomorphism.

Let $X$ and $Y$ be metric spaces, $X$ compact, and $T:X \to Y$ bijective and continuous. Show that $T$ is a homeomorphism. My attempt: We need only show that $T^{-1}$ is continuous. Let $M ...
1
vote
1answer
51 views

Open subsets of $\mathbb{R}^2$

Open subsets of $\mathbb{R}$ can be written as disjoint unions of open intervals- can the same be said in $\mathbb{R^2}$? Open subsets of $\mathbb{R^2}$can be written as disjoint unions of open ...
1
vote
1answer
16 views

Equality between weight and density in metric spaces

I have to prove that in any metric (in generalized version metrizable) space weight of the space is equal to its own density. My job done so far: $$(X,\delta)$$ Is topological space with metric ...
0
votes
1answer
28 views

Why the separate notation for norm

One usually denotes the norm as $\|\cdot\| $, $\| v\| := \sqrt{\langle v, v \rangle}.$ However, in metric spaces, one often writes $d(x,y) \equiv \lvert x-y \rvert$. Since the norm canonically ...
1
vote
0answers
52 views

On a function $f: \mathbb R^m \to \mathbb R^n$ , $n>1$ , mapping connected sets to connedted sets and discontinuous at a point

Let $f: \mathbb R^m \to \mathbb R^n$ be a function mapping connected sets to connected sets where $n>1$ ; let $a \in \mathbb R^m $ and $ \epsilon >0$ be such that $f(B_{\delta}(a)) \cap ...
2
votes
0answers
23 views

Limit and Isolation points.

I have attempted a question that says to prove that the set of isolated points of a countable complete metric space X forms a dense subset of X. My Attempt It has been shown previously that the ...
0
votes
1answer
35 views

Prob. 2, Sec. 20 in Munkres' TOPOLOGY, 2nd ed: Does this metric on $\mathbb{R} \times \mathbb{R}$ induce the dictionary order topology?

Here's Prob. 2, Sec. 20 in the book Topology by James R. Munkres, 2nd edition: Show that $\mathbb{R} \times \mathbb{R}$ in the dictionary order topology is metrizable. This question has ...
0
votes
1answer
25 views

Accumulation point in real spaces

Sequences in $\mathbb{R}^n$ have a unique limit. Is it true that for any sequence which converges to limit exist there exists no accumulation point a such that $x \neq a$. i.e. does unique limit ...
1
vote
1answer
48 views

If $X$ is a non-compact metric space, can $X^n$ ever be compact?

Do there exist metric spaces $X$ such that $X^n$ is compact even though $X$ is not? Since compact spaces can have non-compact subspaces, e.g. $[0,1)\subset[0,1].$
2
votes
1answer
41 views

Showing a sequence converges weakly.

Let $f \in L_2(\mathbb{R})$. How can I show that the sequence ${g_n}$ converges weakly to $0$ in $L_2(\mathbb{R})$, where $g_n(x) = f(x − n)$? If this is not true could someone provide a counter ...
0
votes
1answer
14 views

Almost negative definite matrices and norm-distance matrices

An "almost negative definite" matrix $A$ satisfies the property $$ v^te = 0\implies v^tAv\le 0 $$ where $e=(1,1,\dots,1)$. We know that if $A$ is a simmetric zero-diagonal (hollow) matrix, then $A$ ...
-4
votes
1answer
30 views

Metric Space Topology…/// [closed]

Actually I've no idea how to approach to the correct answer...
0
votes
0answers
13 views

Finite open subsets of a metric space

We know that every open subsets of a finite metric space is finite. Is it possible to have an infinite metric space (not discrete) having a non-empty finite open set? In that case the metric space ...
2
votes
1answer
37 views

Showing $S^2/\sim$ (real projective plane) is Hausdorff

Let $\pi:\;S^2\to S^2/\sim$ be the projection map where the relation on $S^2$ is $a\sim b\iff a =\pm b$. I am trying to show $S^2/\sim$ is Hausdorff. So take $\alpha,\beta\in S^2/\sim$ then ...
0
votes
0answers
23 views

Metrics from Operator Norms

Let $X$ be a Hilbert space and $(\cdot,\cdot)_X$ be the inner product on $X$. It is well known that $|x|_X = \sqrt{(x,x)_X}$ is a norm on $X$ and $|x-y|_X$ is a metric on $X$. The norm on $X$ induces ...
0
votes
0answers
22 views

If $F$ is a compact subset of a metric space $X$, prove $F$ is closed

I was asked to prove this in an exam. I began by proving that $X \setminus F$ is open. I assumed there was an $x\in X\setminus F$ such that $x_k\rightarrow x$. Then draw a ball of radius ...