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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Strong equivalence between Lévy’s metric and a topologically equivalent metric

Let $\mathscr B$ be the Borel $\sigma$-algebra on $\mathbb R$ and let $\mathscr P$ denote the set of all probability measures on the measurable space $(\mathbb R,\mathscr B)$. Lévy’s metric on ...
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Relationship to weak toplogy (Lévy metric)

By $P(\Omega)$, denote the space of all probability measures on $(\mathbb{R},\mathcal{B})$. Let $F_{\mu}$ denote the distribution function of $\mu\in P(\Omega)$. Let, $$ ...
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Prove that a finite set $X$ has exactly one topology that arises from a metric $d$

Given $(X,d)$, I need to show the above statement. I found a question here that I initially thought would answer my query Showing that metric induces single unique topology on a finite set However, ...
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Metric (tensor?) on a cylinder with radius 1 and infinite extent

I have a question and I'm not exactly sure if I'm on the right track. It isn't homework, just a curiosity I'm following: Consider a right circular cylinder with fixed radius of 1. This is ...
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TVS on the reals which or which not induces convergence in norm

Recently I wondered, if convergence in some given metric $d$ on $\mathbb R^n$ induces convergence in norm. Of course, if $d(x,y) = \|f(x)-f(y)\|$, where $f$ is a bijection on $\mathbb R^n$, then this ...
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Discrete metric, singleton open or closed set?

Could someone check the following, is my reasoning correct? EDIT: the following contains errors: see comments Let $$d_\text{disc}(x,y) = \begin{cases}1 & \text{if } x\not = y\\ 0 & ...
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Proving an Ultrametric Space and Its Properties

Let $X$ be the collection of all sequences of positive integers. If $x=(n_j)_{j=1}^\infty$ and $y=(m_j)_{j=1}^\infty$ are two elements of $X$, set $$k(x,y)=\inf\{j:n_j\neq m_j\}$$ and $$d(x,y)= ...
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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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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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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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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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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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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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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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$A \subset \mathbb{R}^n$. If every continuous function $f: A \rightarrow \mathbb{R}$ is is bounded and attains its bounds then A is compact.

I'm doing a metric spaces course and got stuck on proposition. I have a feeling that I want to show that $A$ is bounded and closed then use Heine-Borel theorem. The proposition states that $f$ is ...
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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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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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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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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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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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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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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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Does every manifold M always admit a Riemannian metric?

In the book "Geometry and Topology for Physicists" by Nash and Sen, in Section 7.6, after showing that the structure group $GL(n,\mathbb{R})$ of a frame bundle $F(M)$ (for a general manifold $M$ of ...
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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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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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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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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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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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For any point $a$ of a compact subset $S$ of a metric space, prove that there exists a nearest point $ c $ to $ a $.

Let $S$ be a compact subset of $X$. Define a metric space $(X, p).$ Prove that for any point $a\in X$, there exists a nearest point $c$ in $S$ to $a$. Moreover, $c$ in $S$ such that $p(c,a)\leq ...
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Ultraproduct of a metric space

I am currently trying to understand "Curvature bounded below: a definition a la Berg--Nikolaev" by Nina Lebedeva and Anton Petrunin. They start with a complete, intrinsic metric and space $X$ and say ...
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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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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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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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Length structure topology

Let $(X,T)$ be a Hausdorff topological space. Suppose $\emptyset\neq A\subset C(X)$, where $C(X)$ is the set of curves in $X$, that is, the set of continuous maps $\alpha:I\to X$, where ...
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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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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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Does every non-compact bounded metric space support an equivalent metric in which it is unbounded?

Consider $X$ be an infinite set. Let $d$ be a non compact bounded metric on $X$. Can we define an unbounded metric $d'$ on $X$ such that both the metric spaces $(X,d)$ and $(X,d')$ give the same ...
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Let S be a subspace of topological space X. Show that the closure of S, the set of contact points, is indeed closed.

Let $S$ be a subspace of topological space $X$. Show that the closure of $S$, the set of contact points, is indeed closed. I need to prove that the closure is closed but I don't know how to ...
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Metric assuming the value infinity

If we instead define a metric as $d:X\times X \rightarrow [0,\infty]$, do we lose any nice properties of metric spaces? The reason I ask is that I saw this theorem: Given a finite measure space ...
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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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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: $x \in \overline E$ $x \in \operatorname{Der}(E) = \{x \text{ is an ...
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$\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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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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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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Given two metrics $\rho,\sigma$ on $\chi$ show that the following is a metric on $\chi$

The problem asks us to show that $$ \rho_{2}=(\rho^2+\sigma^2)^{1/2} $$ is also a metric on $\chi$. I'm having trouble showing the triangle inequality; I've tried numerous algebraic manipulations ...
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a dense set in plane

Is there a dense set in $\Bbb{R^2}$ that every vertical line or horizontal line intersect in finite points. I think that we can consider $\Bbb{Q} ×\Bbb{Q}$ but every vertical line or horizontal line ...
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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 ...