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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Show that $\Delta = \{(y,y):y\in N\}\subset N\times N$ is a closed subset of $N\times N$

I have to show that $$\Delta = \{(y,y):y\in N\}\subset N\times N$$ is a closed subset of $N\times N$ I can do this by showing that its complement is an open subset of $N\times N$, but a previous ...
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$M,N$ metric spaces, $\phi:M\to N$ a surjective open map. Show that the map $f:N\to P$ is continuous iff $f\circ \phi$ is continuous

I need to show the following: $M,N$ metric spaces, $\phi:M\to N$ a surjective open map. Show that the map $f:N\to P$ is continuous iff $f\circ \phi$ is continuous In order to show that the composite ...
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boundary of $\{x\in M: f(x)>0\}\implies f(x)=0$

Given a continuous function $f:M\to \mathbb{R}$, and $A=\{x\in M: f(x)>0\}$, I need to show that if $x\in \partial A$(boundary of $A$), then $f(x) = 0$. I know that $\partial A$ is the set of all ...
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26 views

Relation between Compactness, Closedness and Completness of metric spaces

I would like to know as many relations as possible to get a better picture. I know that if $f$ is continuous and $(X,d)$ is complete, then $f(X)$ is complete $\iff$ closed. Question:However, are ...
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28 views

Separable metric spaces that are not normable

Not quite sure whether this question belongs here or on MESE. Anyway: Can anyone suggest a good example of a separable metric space that is neither normable nor a subset of normed space with the ...
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16 views

In search of a necessary condition for completeness of some metric space with application to pde

$A$ is an operator. Consider a metric space $K$ (a function $f$ is in $K$ if and only if $Af$ is in $L^2$) where the metric between two functions $f$ and $g$ is defined as $\mu (f ,g) = \int_{R^3} ...
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18 views

Kantorovich-Rubinstein Theorem - lower semi-continuity of distance function

For instance in "Topics in Optimal Transportation" by Cédric Villani, it is claimed that the 1-Wasserstein-distance of two measures $\mu, \nu$ on a certain space $X$ can be expressed as the supremum ...
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1answer
27 views

Countable Metric Space and nowhere dense sets

Hey I could not find an answer on wether or not a countable Union of nowhere dense sets is nowhere dense in a countable metric space. I know that a countable Union of nowhere dense sets is not always ...
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11 views

Modulus of continuity example

could help me clear something out. I am looking for an example of modulus of continiuity. I was told that is doable for: $f(x)=\sqrt{1-x^2}$ for $x\in [0,1]$. Some calc: From ...
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Coproduct in the category of metric spaces

While discussing categories without coproducts, we stumbled with the category $\mathbf{Met}$ that takes metric spaces as its objects and short maps as its morphisms. It is claimed that $\mathbf{Met}$ ...
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Show $h(x)=1+4d(x,x_0)$ is continuous in a metric space $(X,d)$ and $(\mathbb R , |\cdot |) $

Let $(X,d)$ and $(\mathbb R , |\cdot |) $ be metric spaces. Show that $h(x)=1+4d(x,x_0)$ is continuous. I thing that a composition of continuous functions is continuous, but I don't think I ...
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22 views

Interior and closure of $l^1 (\mathbb{N})$ in $ l^{\infty}$?

Let $$ l^{\infty} = \left\{ (x_n)_{n \in \mathbb{N}} \in \mathbb{R}^{\mathbb{N}} \mid (x_n)_{n \in \mathbb{N}} \ \text{is bounded} \right\} $$ Here $\mathbb{R}^{\mathbb{N}}$ denotes the space of all ...
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$f:\mathbb R^2 \to \mathbb R$ be a function , $|f(x)-f(y)|\ge 3||x-y|| , \forall x,y \in \mathbb R^2$ ; is $f(\mathbb R^2)$ open in $\mathbb R$?

Let $f:\mathbb R^2 \to \mathbb R$ be a function such that $|f(x)-f(y)|\ge 3||x-y|| , \forall x,y \in \mathbb R^2$ , then is it true that $f$ maps open sets of $\mathbb R^2$ to open sets of $\mathbb R$ ...
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33 views

Show that the projection $M_1\times \cdots \times M_n\to M_i$ given by $p_i((x_1,\cdots,x_i,\cdots,x_n)) = x_i$ is a continuous and open map.

I have the following question: Let $M_i$ be metric spaces. Show that the projection $M_1\times \cdots \times M_n\to M_i$ given by $p_i((x_1,\cdots,x_i,\cdots,x_n)) = x_i$ is a continuous and open ...
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20 views

Contraction mapping with no fixed point using a incomplete metric space

I know that if $f:X\rightarrow X$ is a contraction, then $d(f(x),f(y))\leq \alpha d(x,y)$ for $0<\alpha<1$. I'm looking for a counter example, that is a metric space that's incomplete, and ...
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What is the Most General Setting in Which Limits Commute with Continuous Functions?

In metric spaces, we have that limits commute with continuous functions. In Hausdorff spaces, the limits of nets are always unique. Seemingly the second fact is necessary for the proof of the ...
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19 views

$f:M\to N\times N$ is continuous and $\Delta = \{(y,y):y\in N\}\subset N\times N$ then $f^{-1}(N\times N-\Delta)$ is an union of open balls in $M$

I need to show the following: $f:M\to N\times N$ is continuous and $\Delta = \{(y,y):y\in N\}\subset N\times N$ then $f^{-1}(N\times N-\Delta)$ is an union of open balls in $M$ But I have no idea of ...
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$A$ bounded implies $f(A)$ bounded? (if $f$ is continuous)

if $A$ is bounded and $f$ is continuous, then I know $f(A)$ is bounded... But I just can't picture it. I keep thinking of this scenario: $f = \frac 1x $ is continuous (at least on $(0, \infty)$ ) ...
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34 views

Is every complete metric space closed?

I know that if $A\subset X$ where $X$ is a complede metric space, and $A$ is closed $\iff$ it's complete. However is every metric space closed? E.g., can I take $X\subset X$ and since $X$ is ...
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Confirming the triangular inequality for Lévy-metric $d_L(F,G)$ and $d_L(F,G)<\infty$

Let $F,G$ be cumulative distribution functions. The Lévy-metric is defined to be $$ d_L(F,G)=\inf\left\{h\geq 0: F(x-h)-h)\leq G(x)\leq F(x+h)+h~\forall x\in\mathbb{R}\right\} $$ I would like to ...
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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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44 views

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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12 views

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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19 views

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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26 views

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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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 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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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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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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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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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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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 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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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 $\rho_{2}(x,y) = \displaystyle \frac{\rho_{1}(x,y)}{1+\rho_{1}(x,y)}$. I need to ...
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24 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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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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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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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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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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305 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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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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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
40 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 $\{0,1\}$. ...
2
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1answer
28 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
15 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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1answer
41 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
68 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 ...