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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Pseudometric in Minkowski space obeys the triangle inequaltity if events are spacelike separated

Define the following Pseudo-metric on Minkowski space-time $$d(X,Y)=\sqrt{|\eta([X-Y],[X-Y])|}$$ Where $\eta(X,X)=X^T\eta X$ and $$\eta=\operatorname{diag}(-1,1,1,1)$$ the diagonal matrix with those ...
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24 views

Show a set is open using open balls

The set is $ \{ (x_1 , x_2) : x_1 + x_2 > 0 \}$ I wanted to solve this using open balls, so I said let $y = (y_1, y_2)$ be in the stated set. Then create an open ball $ B_r (y)$ around this ...
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1answer
17 views

Prove a cauchy sequence in $(X,\rho)$ maps to a cauchy sequence in $(Y,\sigma)$

Let $(X,\rho)$ and $(Y,\sigma)$ be two metric spaces. Assume ${x_n}$ is Cauchy in $X$, and that $f:X \rightarrow Y$ is uniformly continuous. Prove that $f(x_n)$ is Cauchy in $Y$. Take ...
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2answers
43 views

Prove the mapping is continuous

$f(x):=\rho(x,T(x))$ where $T(x)$ is a Lipschitz function. $(X,\rho)$ is a compact metric space and $T:X\rightarrow X$. I need to prove $f(x)$ is continuous. I'm trying to use the triangle inequality ...
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1answer
28 views

Disconnected metric space and continuous functions

Question: Give an example of disconnected metric space $X$ and a metric space $Y$ such that for every continuous function $f: X \to Y$, $f(X)$ is a connected subset of $Y$. I was thinking about ...
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1answer
24 views

Show that $d( \; \cdot \; ,A)$ 1-Lipschitz continuous

Let $(X, d)$ be a metric space and $A\subset X$, with $$d(x,A) = \inf_{y \in A} d(x,y)$$ Now my problem is to show the following:$$ \forall_{x,z \in X}\mid d(x,A)-d(z,A)| \le |x-z| \;\; \text{(which ...
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1answer
32 views

Showing that a family of metrics induce all the same topology on special sequence space

Let $X = \{0,1\}$ and consider the discrete metric $$ d(x,y) := \left\{ \begin{array}{ll} 0 & x = y \\ 1 & x \ne y. \end{array}\right. $$ Now consider $X^{\mathbb N_0}$, the set of all ...
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1answer
21 views

If every borel measurable function continuous in compact metric space then metric space is finite

Let $(X,d)$ be a compact metric space. Suppose every Borel measurable function $f : X \to \mathbf{R}$ is also continuous. Show that X is a finite set. Thank you for your time
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Give an example of compact set in some metric space $X$ such that $X$ is neither $\mathbb{R}^n$ nor a finite set. [on hold]

Or at least an example of a metric space $X$ such that $X$ is neither $\mathbb{R}^n$ nor a finite set.
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17 views

Prove that the union of two given subsets of $\Bbb{C}^n$ is path-connected

Consider a subset $A$ of $Z=(\Bbb{C}^n$, Zariski topology) and regard it as a subspace of ($\Bbb{C}^n$, Metric topology). Sine $\Bbb{C}^n$ is homeomorphic to $\Bbb{R}^2n$, we can decide if A is ...
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1answer
7 views

Prove that that a map in $C^N$ in the metric topology is continuous to the Zariski topology, but that the map is not a homeomorphism.

Prove that that a map in $C^N$ in the metric topology is continuous to the Zariski topology, but that the map is not a homeomorphism. My Work: I plan to use the fact that the metric topology is a ...
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1answer
25 views

Show that two metrics known not to be strongly equivalent actually induce the same topology.

Suppose on $\mathbb{R}$, we have the usual Euclidean metric, $\rho_{1}(x,y) = \Vert x-y \Vert$, and also the metric $\rho_{2} = \displaystyle \frac{\rho_{1}(x,y)}{1+\rho_{1}}$. I need to show that ...
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2answers
22 views

does the condition “every open set is a countable union of closed sets” imply metrizability

In metric spaces, every open set is a countable union of closed sets. is the converse true? A topological space with the property "every open set is a countable union of closed sets" has to be ...
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24 views

Normed space, inner product space, which is larger?

My textbook says normed space is inner product space unless it satisfies some parallelogram identity. In this case, we can conclude inner product space is a subset of normed space. However, norm is ...
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38 views

Metric space on $\mathbb{R^n}$ where Heine-Borel criterion does not hold

Heine-Borel criterion of $\mathbb{R^n}$ : closed and bounded $\implies$ compactness Give an example of a metric space in $\mathbb{R^n}$ where this criterion does not characterize compactness ...
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22 views

On lifts of a trajectory of a quadratic differential

Let $X$ be a Riemann surface of genus $g\ge 2$ and $q$ a holomorphic quadratic differential on $X$. The differential $q$ defines a flat metric with conical singularities on $X$: if $q=f(z)dz^2$ ...
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1answer
23 views

Does continuity in $(X, d_X)$ imply continuity in $(Y, d_Y)$ when $(X, d_X) \simeq (Y, d_Y)$?

I want to check if my intuition about continuity is correct. Suppose $(X, d_X)$ and $(Y, d_Y)$ are two metric spaces that are isometrically isomorphic, i.e., there is an isomorphism $h : X \to Y$ ...
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2answers
28 views

A metric space (X,d) in which any intersection of open sets is open

Assume we have a metric space (X,d) that satisfies the condition that the intersection of any collection of open sets is open. Explain which subsets of (X,d) are open?
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1answer
21 views

If $\{u_n\} \to u$ and $\{v_n\} \to v$. Show that $\{\rho(u_n, v_n)\} \to \rho(u,v)$

Let $(X,\rho)$ to be a metric space in which $\{u_n\} \to u$ and $\{v_n\} \to v$. Show that $\{\rho(u_n, v_n)\} \to \rho(u,v)$ Proof: Suppose $\{u_n\} \to u$ and $\{v_n\} \to v$. This means that ...
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2answers
303 views

Distance from a compact subset need not be attained in a metric space?

Suppose we have a metric space $(X,d)$. Let $S$ be a compact subset of $X$. Provide me with an example of $X$, and $S$ (closed and bounded in $X$) such that $$\min \{d(p_0,p): p \in S\}$$ does not ...
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15 views

How does one get $p=2$ from a condition that there be non-trivial linear transformations of every dimension that to any power are $p$-norm-preserving?

Verifying that (p=2) satisfies $$\forall n\in\mathbb{Z}^+.\exists A\in(\mathbb{R}^{n\times n}\setminus\{I_n\}).\forall k\in\mathbb{R}.\forall ...
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1answer
29 views

Any subset of a metric space is an infinite union of some individual elements of the space?

Let $E$ be a metric space such that the set $\{x\}$ is open $ \forall x \in E$. Does the following proposition make sense? All subsets of $E$ are open. Proof: $\forall S \subset E$, there are ...
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2answers
28 views

Does every metric on a non empty set can be extended on a super set to a metric?

Let $\phi \ne X \subseteq Y$ , let $d$ be a metric on $X$ , then does there exist a metric $d'$ on $Y$ such that $d(x,y)=d'(x,y) , \forall x, y \in X$ ? What if we also assume that the metric $d$ on ...
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3answers
38 views

What is the logic underlying this proof?

Proposition: A metric space $X$ is connected if, and only if, every continuous function $f:X\to (\{0,1\},d_D)$ is a constant function, where $d_D$ is the discrete metric on the set $\{0,1\}$. ...
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1answer
24 views

Prove that two metrics are equivalent

I got stuck on this problem. Hope someone can give some hint to move on. Thanks. Suppose $d_1(x,y) = |x-y|$, $d_2(x,y)=|\phi(x) - \phi(y)|$ where $\phi(x) = {x \over {1 + |x|}}$. Prove that $d_1$ ...
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1answer
14 views

If an open neighborhood of $x$ has infinite points of $E$, then $x$ is a limit point of $E$

Let $(X, d)$ be a metric space, $E \subseteq X$ and $x \in X \setminus E$. Prove that the following are equivalent: $x \in \overline E$ $x \in \operatorname{Der}(E) = \{x \text{ is an ...
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37 views

$\mathbb{R}^2$ to $\mathbb{R}^1$ Injective Mapping While Preserving the Triangle Inequality

Is there a way to map from $\mathbb{R}^2$ to $\mathbb{R}^1$, such that every point in $\mathbb{R}^2$ has a unique point in $\mathbb{R}^1$ and you preserve the distance (isometry) relations of ...
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1answer
65 views

Topological spaces without homeomorphisms?

Can we find a topological space which is not homeomorphic to any other? Of course, not considering the space itself neither the empty set. And if's so, is it possible to classify them? Just like the ...
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1answer
18 views

Proper map and sequences in metric spaces

Let $f:X\to Y$ be a continuous map between metric spaces satisfying the Heine-Borel theorem. Show that $f$ is proper if the following condition holds: For every sequence $x_n\in X$ such that ...
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1answer
32 views

Proving that if $d(x, a) < \varepsilon$ for every $a \in A$, then $d(x, b) \geq \varepsilon$ for every $b \in X \setminus A$

I want to prove the following result: Let $(X, d)$ be a metric space. Then $$\mathring E = \{x \in X \mid d(x, X \setminus E) > 0\}$$ where $d(x, A) = \inf\limits_{y \in A} d(x, y)$. This ...
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Error in proof that the closure of open ball equal the closed ball in all metric spaces

Let $(X, d)$ be a metric space. Denote the open and closed ball as $$B(x_0, r) = \{x \in X \mid d(x, x_0) \lt r\},$$ $$D(x_0, r) = \{x \in X \mid d(x, x_0) \leq r\}.$$ Then $\overline{B(x_0, ...
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1answer
24 views

How to show that all sequentially compact spaces are bounded?

I want to show given a sequentially compact subset $A \subseteq M \implies A$ is bounded. I read this Every sequentially compact set is closed and bounded. but the proof is poorly written and jumpy ...
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Constructing a metric $\rho$ such that $(\mathbb{R}\setminus \{0\},\rho)$ is a complete metric space

Let $S = \mathbb{R}\setminus \{0\}.$ Construct a metric $\rho$ on $S$ such that (1) $(S,\rho)$ is a complete metric space and (2) for any sequence $\{s_n\}$ in $S$ and $s \in S,$ the ...
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Root distance function in Metric space [duplicate]

Let $\mathbf X = \Bbb R$ with distance function defined by $d(x,y) = {|x-y|}^\alpha$ , where $\alpha \in \Bbb R$ $(0<\alpha\le1)$. Prove that $(\Bbb R , d)$ is a metric space. The first three ...
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1answer
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Is the space of subsets of $\mathbb R^n$ with the Hausdorff metric separable?

Let M be the Metric Space whose "points" are the Closed and Bounded subsets of a finite dimensional Euclidean Space and whose "distance function" is the Metric defined by Hausdorff for such point ...
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1answer
22 views

Show that $d_2$ is not a metric.

Show that the function $d_2$ given by $d_2(f_1, f_2)^2 = \int_a^b{(f_1 - f_2)^2}$ is not a metric space on the space of Riemann integrable functions on $[a,b]$. $d_2(f_1, f_2) = 0$ iff $f_1 = ...
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1answer
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Should a metric always map into $\mathbf{R}$?

Typically you see the definition of a metric as a function which maps $X\times X\to\mathbf{R},$ but does this always have to be the case? Motivating example: When you complete $\mathbf{Q}$ with the ...
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simple proof for principle of pigeons

I must prove the principle of pigeons but the proofs I find in the internet are too complex. Here's what I can use: Definition $$I_n = \{p\in \mathbb{N}; p\le n\}$$ The principle of the pigeons ...
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Example of a bounded space which is not totally bounded

I was trying to find an example of a bounded metric space which is not totally bounded. The only example I could come up whith was the natural numbers with the discrete metric. However, like any ...
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1answer
14 views

Dense subset in which Cauchy sequences are convergent

Let $S$ be a dense set of a metric space $X$, such that all Cauchy sequences in $S$ are convergent (not necessarily in $S$). Then $X$ is complete space. How can I show that $X$ is complete space ...
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21 views

Is this orthogonal distance a common pseudometric?

Define $d: V \times W \to \mathbb{R}$ such that $$d(v,w) = \sup_{z \perp w} \frac{\langle z, v \rangle}{\|v\|\|z\|}.$$ Is this a pseudometric that anyone has utilized in the literature? Does it have a ...
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20 views

distribute K points in N dimensional space

I'll try to do my best to simplify the problem, I'm not a Mathematician, I'm a Computer Engineering Student. I'm doing the K-means algorythm, for those who doesn't know what is, is an algorythm to ...
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1answer
22 views

If $F$ is closed subset of $R^n$ and $x \in R^n, $ is $x+F$ still closed? [closed]

If $F$ is closed subset of $R^n$ and $x \in R^n, $ is $x+F$ still closed ?
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1answer
17 views

Prove directly from definition: countably compact subsets of metric spaces are closed

I am trying to prove the statement that every countably compact subset Y of a metric space (X,d) is closed. I am aware of the fact that, for metric spaces, countable compactness is equivalent to ...
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1answer
20 views

Puzzled with this number theory/analysis problem

So, I am having this problem, let $N(x,y)$ be the greatest integer which $b^{N(x,y)}|x-y$ where $x,y$ are integers in $\mathbb{Z}$. Assume that $b \geq 2$. Show $d(x,y)=b^{-N(x,y)}$ is a metric. ...
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Is a ball always connected in a connected metric space?

If I have a connected metric space $X$, is any ball around a point $x\in X$ also connected?
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33 views

Cauchy sequence of natural numbers

Consider the set consisting of all cauchy sequences $a_n$ with $a_n \epsilon \mathbb{N}$ for all $n$. Is the set countable? My idea: It is straight forward to prove that any such cauchy sequence ...
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2answers
71 views

Does there exist a Vector that can't be written as a Tuple of Scalars?

The most abstract/general definition of a vector The most general definition of a vector is as an element of a vector space. Given a vector $u$, we can always say that there exists a vector space $V$ ...
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1answer
25 views

Function of metric with a fixed point

I'm trying to prove that given a metric space $(X, d)$, for a fixed $x\in X$, define the function $g(y)=d(x,y)$, then $g(y)$ is continuous, using triangle inequality. My first question is that can I ...
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1answer
19 views

Isometries in $\mathbb{E}^2$

We define $\mathbb{E}^2 = (\mathbb{R}^2, d_E)$, where $d_E$ is the usual Euclidean metric on $\mathbb{R}^2$. We say that a function $f:X \rightarrow Y$, where $X, Y$ are metric spaces, is an isometry ...