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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Proving this function is continuous and bounded

Let $(X,d)$ be a metric space. Fix $a \in X$. For every $x \in X$ we consider the function $$f_x : X \rightarrow \mathbb{R}: y \mapsto f_x(y) = d(x,y) - d(a,y). $$ Prove that $f_x$ is continuous ...
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In what sense is metric space completion universal?

The completion of a metric space is unique up to metric monomorphism (usually called isometry). It is also the "obvious" way to make all Cauchy sequences convergent. Structures which are unique up ...
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I am searching for a metric space in which every Cauchy sequence is stationary.

I am searching for an example of a metric space in which every Cauchy sequence is stationary, means that there exists some $N$ such that for all $n>N$: $a_n = a_N$. Is there a simple example for ...
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surface to surface map, $f$ is closed but neither open nor continuous

I'm trying to teach my self topology. The book I'm using has the following problem: Give an example of two subsets $X,Y \subseteq \mathbb R ^2$, both considered as topological spaces with their ...
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80 views

Proving triangle inequality using complete-linkage between clusters and arbitrary dissimilarity measure

Assuming a dissimilarity measure d satisfies the usual properties, I need to prove that complete linkage ( i.e. d(A,B)=maxx∈A,y∈B{d(x,y)} ) either satisfies or does not satisfy the triangle inequality ...
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General structure of the proof that every compact metric space is the continuous image of the Cantor set

I am reading the book "General Topology" by Stephen Willard and I have the theorem "Every compact metric space is a continuous image of the Cantor set". The book also presents its proof. The proof ...
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How do you find the metric tensor for a given manifold?

Is there some general way to derive the metric tensor for a given manifold M? For example, how was the metric for the surface of a sphere $$ds^2=d\theta^2+\sin^2\theta \, d\phi^2$$ first derived?
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Proving $(C_b(X,Y),d_{\infty})$ is complete if $(Y,d_y)$ is complete

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. We say a function $f: X \rightarrow Y$ is bounded if $f(X)$ is a bounded set of $Y$. Consider $$C_b(X,Y) = \left\{f: X \rightarrow Y \mid f \ \text{is ...
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Showing $(l^1(\mathbb{N}), d_1)$ is complete

Let $$ l^1(\mathbb{N}) = \left\{(x_n)_n \mid \sum_{n=0}^{\infty} |x_n| \ \text{converges} \right\}. $$ Prove that $(l^1(\mathbb{N}), d_1)$ with $d_1(x,y) = \sum_{n=0}^{\infty} |x_n - y_n|$ is a ...
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Every metric $\delta$ on sets is $\delta (A, B) = \mu(A \Delta B)$, resp. $\delta(A, B) = \mu(A \Delta B)/\mu (A \cup B)$, for some measure $\mu$?

Suppose $\mu$ is a measure and that $\delta (A, B) = \mu(A \Delta B)$ (where $\Delta$ represents symmetric difference, and $A=B$ whenever $\mu(A \Delta B) = 0$). Then $\delta$ is a distance function. ...
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How do I prove that the compact open topology is metrizable?

Reference: Conway - Functions of one complex variable Let $G$ be open in $\mathbb{C}$ and $\{K_n\}$ be a sequence of compact subsets of $G$ such that $\bigcup_n K_n = G$ and $K_{n}\subset Int(K_{n+...
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Jaccard dissimilarity and the triangle inequality

Suppose that $\delta(A, B) = \dfrac{A \Delta B}{A \cup B}$, where $\Delta$ represents symmetric difference. Then how does one prove the triangle inequality, viz that $\delta(A, B) + \delta(B, C) \ge \...
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Uniformly Cauchy sequence of functions

I am trying to show the following: For each $n \in \mathbb N$, let $f_n:X \to Y$, where $(Y,d)$ is a complete metric. Suppose that for every $\epsilon>0$, there exists $n_0 \in \mathbb N$ such ...
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Boundary of $M_rp$ not equal to the sphere of radius $r$ at $p$?

My problem is to find a metric space in which the boundary of $M_rp$, where $M_rp = \{q \in M: d(p, q) < r \}$, is not equal to the sphere of radius $r$ at $p, \{x \in M: d(x, p) = r\}$. ...
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Showing $f(a) \in V$ but $f(x_n) \notin V$ for every $n$

Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces and let $f: X \rightarrow Y$ be a function. Let $a \in X$ and suppose $f$ is not continuous in $a$. Prove that there exists an open subset $V$ in $Y$ ...
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$A,B$ be countable dense subsets of $\mathbb R$ , let $A,B$ be given usual subspace topologies , then there exists a homeomorphism $f:A \to B$?

Let $A,B$ be countable dense subsets of $\mathbb R$ (with usual euclidean topology ) let $A,B$ be given usual subspace topologies , then is it true that there exists a homeomorphism $f:A \to B$ ?
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Is this metric space normable?

Given the function $\rho:\mathbb{R}^n\to\mathbb{R}_+$ defined by $$\rho(x)=\log\left(\frac{\sum\left|x_i\right| e^{\left|x_i\right|}}{W\left(\sum\left|x_i\right| e^{\left|x_i\right|}\right)}\right)$$ ...
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Space of continuous functions with compact support dense in space of continuous functions vanishing at infinity [on hold]

How can we prove that the space of continuous functions with compact support is dense in the space of continuous functions that vanish at infinity?
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Show that anti-metric space can only have one point

Let's define new object. Given $X$ a set: Let anti-metric be defined as: $b: X\times X \to \mathbb{R}$ such that: $b(x,y)\ge 0, \thinspace \forall x,y \in X$ $b(x,y)=0\Rightarrow x=y$ $b(x,y) = b(y,...
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Urysohn Metrization Theorem contradiction (uniform topology homeomorphic to product topology)?

The theorem states that if $F$ is regular and has a countable basis, then it is metrizable. In Munkres' proof of this theorem, he gives a function (homeomorphism) $F:X \rightarrow [0,1]^\omega$ that ...
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Can a metric space over integers induce a topology?

Questions to get a better grasp of basic topology: A metric space is an ordered pair $(M,d)$ where $M$ is a set and $d$ is a metric on $M$, i.e., a function $$ d \colon M \times M \to \...
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What is the motivation of Levy-Prokhorov metric?

From Wikipedia Let $(M, d)$ be a metric space with its Borel sigma algebra $\mathcal{B} (M)$. Let $\mathcal{P} (M)$ denote the collection of all probability measures on the measurable space $(...
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Showing this set $A$ is closed, bounded and not compact?

Let $$ l^1(\mathbb{N}) = \left\{ (x_n)_n \mid \sum_{n = 0}^{\infty} | x_n | \ \text{converges} \right\}, $$ the space of all sequences whose associated series converge absolutely. On this space we ...
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Generalizing norms: leaving out absolute homogeneity

Given a function $\rho:X\to\mathbb{R}$ on a vector space $X$ which satisfies the following properties: $\rho(x)=0$ if and only if $x=0$ $\rho(x+y)\leq\rho(x)+\rho(y)$ $\rho(-x)=\rho(x)$ for any $...
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Alternative proof: show that any metrizable space $X$ is normal - Part 1

There is a proof online that shows that all metric spaces are normal. The proof is as follows However, it has the additional baggage of needing to show that $d(x,A)$ is continuous and $U,V$ are ...
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Uniform convergence of a sequence of functions and equicontinuity

Let $(X,d)$ be a compact metric space. I would like to prove that if $(f_n)_{n \in \mathbb{N}}$ is a sequence of continuos functions $f_n:X \to Y$ that converge uniformly in $X$, then $(f_n)_{n \in \...
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Give example of $f$ that is open but neither closed not continuous (in 2D).

I'm trying to teach my self topology. The book I'm using has the following problem: Give an example of two subsets $X,Y \subseteq \mathbb R ^2$, both considered as topological spaces with their ...
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What subbase generates metric topology?

Let metric topology be the topology generated by metric balls of a metrizable space $X$ Is there a subbase $S$ that generates the metric topology? I am asking because in most textbooks, it seems ...
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Pearson correlation and metric properties

Assuming that the data set was $z$-standardized to zero mean and unit variance (also assuming that it does not contain constant vectors). Then Pearson's r reduces to Covariance: $$\rho(X,Y) := \frac{...
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Explicit construction of an $\epsilon$ net covering

Suppose $X$ is a compact space. In particular $X$ is totally bounded and there exists $x_1,..,x_n$ such that $$ X = \bigcup_{i=1}^n U(x_i, \epsilon) $$ where $U$ is the Open Ball centered at $x_i$ ...
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Proving $\mathbb{R}^2$ is not separable for this metric?

Let $d_S$ be a metric on $\mathbb{R}^2$ defined as follows $$ d_S(x,y) = \begin{cases} || x- y|| & \text{when} \ x \ \text{and} \ y \ \text{are linearly dependent} \\ ||x|| + || y || & \text{...
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Is the following metric topological equivalent to Euclidean metric?

Let $d_S$ be a metric on $\mathbb{R}^p$ defined as $$ d_S(x,y) = \begin{cases} || x- y|| & \text{when} \ x \ \text{and} \ y \ \text{are linearly dependent} \\ ||x|| + || y || & \text{when}\ ...
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Finding a special subsequence of any Cauchy sequence

Let $(X,d)$ be a metric space and let $(x_n)$ be a Cauchy sequence in $X$. Let $(\epsilon_n)$ be a sequence of real numbers and decrease to $0$. Show that there is a subsequence $(x_{n_k})$ of $(x_n)$ ...
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Show $d_2(f,g) = \sqrt{\int\limits_0^1 |f(x) - g(x)|^2 dx}$ is a metric on $C[0,1]$

I am surprised that this question hasn't been asked on here I need to show that $$d_2(f,g) = \sqrt{\int\limits_0^1 |f(x) - g(x)|^2 dx}$$ is a metric on $C[0,1]$ Proof: As usual, positive ...
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Convergence of finite metric spaces to an infinite one

Let $\{(M_i, d_i)\}$ be an infinite sequence of finite metric spaces, where $|M_i|$ is strictly increasing with $i$. Is there a standard definition of what it means for the sequence $\{(M_i, d_i)\}$ ...
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A set $A \subset l_1$ is compact if and only if closed, bounded, and one other condition

A set $A \subset \ell_1$ is compact if and only if $A$ is closed and bounded and given any $\epsilon >0$, there exists $n_0$ such that $\sum_{k=n}^{\infty} |x_k| < \epsilon$ for all $n> n_0$ ...
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In a metric space $X$, a subset $S \subset X$ is relatively sequentially compact if and only if its closure $\overline{S}$ is sequentially compact.

Prove that in a metric space $X$, a subset $S \subset X$ is relatively sequentially compact if and only if its closure $\overline{S}$ is sequentially compact. The terms relatively sequentially ...
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Alternative characterization of complete metric space

Let $(X,d)$ be a metric space. It is complete if every Cauchy sequence for $d$ on $X$ is convergent. I've heard an alternative definition of completeness for $(X,d)$: it is complete iff the ...
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Show that any metrizable space $X$ is Hausdorff

I wish to show that any metrizable space $(X,\mathcal{T})$ is Hausdorff Proof attempt: Let $d$ be the metric that generates the topology on $X$. Pick two points $x,y \in X$, we wish to produce two ...
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Show that any metrizable space $X$ is regular

This is a quick follow up to another question Show that any metrizable space $X$ is Hausdorff Recall, a topological space $(X,\mathcal{T})$ is regular if we can separate any point $x$ from ...
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Can there be a metric space where no contraction has a fixed point?

We know that: If $X$ is a metric space, then every contraction has at most one fixed point. (Note: if metric space is complete, then we have existence and uniqueness) I wonder if there can be a ...
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Show that linear functional $L(f) = \int_0^1 f(x) dx$ is continuous

Let $(C[0,1], d_1)$ be a metric space of all continuous functions $f:[0,1] \to \mathbb{R}$, $d_1$ is the $L_1$ metric $$d_1(f,g) = \int\limits_0^1 |f(x) - g(x)| dx$$ Show that linear functional $L(...
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Locus in $(\mathrm R ,d_{\infty})$ with $d_\infty(x,y)=\max\limits_i |x_i-y_i|$ [closed]

Find the locus of points $(x_1,x_2)$ in the plane such that their distance from $(1,2)$ is equal to $3$ at $(\mathrm R ,d_\infty(x,y)=\max\limits_i |x_i-y_i|)$ I have no idea how this is looks ...
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Proof that the nowhere differentiable functions are dense in $C_b(\mathbb R)$.

I tried to make a proof, where I use a Weierstrass function. I was surprised at how easy it was, and thus a little doubtful as to the correctness of the proof. I've looked it over, and didn't find any ...
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Show that if $(X, \mathcal{T}), (Y, \mathcal{J})$ are both metrizable, then $X \times Y$ with product topology is metrizable

Let $(X, \mathcal{T}), (Y, \mathcal{J})$ be both metrizable, then Claim: $X \times Y$ with product topology is metrizable I have made an attempt at this question but the notation is ...
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Determining if this mapping is continuous?

Let $X$ be a closed and bounded subset of $\mathbb{R}^p$ and let $C(X)$ denote the vector space of continuous functions from $X$ to $\mathbb{R}$. For $f,g \in C(X)$, let $$ d_{\infty} (f,g) = \sup \...
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The set of all polynoms are closed at $d(x,y)=\max\limits_{[a,b]} \mid x(t)-y(t)\mid$?

Prove or disprove with counter-example: the set of all polynoms are closed at $d(x,y)=\max\limits_{[a,b]} \mid x(t)-y(t)\mid$ The polynoms in the interval $[a,b]$ Attempt: counter-example: $y(...
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Are symmetric and $\Delta$-metric common terminologies?

In these notes on metric spaces, the author also defined something known as "symmetric", and $\Delta$-metric. I have never seen these terminologies before. Are these terms standard usage? Can ...
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$X$ be real i.p.s. dim.>1 , if two closed balls,none of which is a subset of the other,intersect then do the boundaries of the balls intersect too?

Let $X$ be a real inner product space of dimension more than $1$ , let $B[x;r] , B[y;s]$ be two closed balls having non-empty intersection where none of the balls is a subset of the other , then is ...
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$f: \mathbb{R}^{n+1}-\{0\}\rightarrow S^{n}$ via $x \mapsto\frac{x}{\|x\|}$ is continuous

I'm having trouble understanding why a map $f: \mathbb{R}^{n+1}-\{0\}\rightarrow S^{n}$ (unit $n$-sphere, $n\ge 1$) via $x \mapsto\frac{x}{||x\|}$ is continuous. Since Unit n-Sphere under Euclidean ...