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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Linear bound on angles in an euclidean triangle.

I am trying to understand a proof in the book of Burago "A Course in metric geometry" (Lemma 10.8.13 page 383). I have difficulties with a certain inequality for the angle of euclidean triangles: ...
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Topology; Definition of the open ball and open sets confuses me

I just picked up T.W Gamelin’s book on topology. I started reading and got confused when I came to the definition of an open ball on the second page. It says $B(x;r) =$ All $y$ in the set $X$ such ...
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Continuous piecewise smooth curve

I cannot understand the definition of $\tilde d(p_1,p_2)$ here? Can anyone please explain it clearly?
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Subbases and half-planes

If $(X,d)$ is a metric space, it's easy to show that $H(x,y)=\{w\in X\mid d(x,w)>d(y,w)\}$ is open in the topology $\tau$ induced by $d$. Is, in general, $\{H(x,y)\mid (x,y)\in X\times X, x\neq y\}$...
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At what conditions a compact metric space of covering dimension $n$ (on $\mathbb R^n$) is an n-manifold?

In the discussion to the MSE post an answer of @MatthewPancia with correction in a comment of @JasonDeVito would state: Every compact metric space of covering dimension $n$ can be embedded ...
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Why is boundedness necessary in this proof that addition and scalar multiplication are continuous maps in the metric space induced by a norm?

Why is boundedness necessary in this proof that addition and scalar multiplication are continuous maps in the metric space induced by a norm? Suppose that $\lambda _n \to \lambda $, $\mu _n \to \mu $,...
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How show the map $f:\mathbb R^2\rightarrow\mathbb R$, defined as $f(x,y)=x+y$ is continuous for all $(x,y)\in\mathbb R^2$?

Question: I want to show the map $f:\mathbb R^2\rightarrow\mathbb R$, defined as $f(x,y)=x+y$ is continuous for all $(x,y)\in\mathbb R^2$. Issue: I know how to prove this via the epsilon-delta way. I ...
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A question about a perfect space and a linear order on it

Suppose I have a nonempty perfect Polish space $X$, and there's some linear order $<$ on it (it is not related to the topology on $X$ in any way). How can I prove that there is a point $y$ in $X$ ...
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Proving the existence of a sequence of polynomials convergent to a continuous function $f$.

I need to show that if $f$ is continuous function ($f:\mathbb{R}\rightarrow \mathbb{R}$), then there exists a sequence of polynomials which converges to $f$ on any compact subset of $\mathbb{R}$. I ...
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A geodesic metric space is a manifold on its own right. What are conditions for a Finsler space to be a manifold?

A geodesic metric space can locally be approximated by a vector space. This approximation provides it with a natural manifold structure. It means that geodesic metric space is more fundamental concept ...
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Is this a metric on matrices?

In the set of $n$-by-$n$ reversible real matrices, decide whether $$d(A,B)=\ln (\lVert A^{-1}B\rVert\cdot\lVert B^{-1}A\rVert)$$ defines a metric and/or semi-metric. Can you please help me to solve ...
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Example of equivalent but not strongly equivalent metrics

Please could someone show me an example of metrics $d$ and $d'$ that are not strongly equivalent but are equivalent? I read the Wikipedia article here but couldn't find an example. For completeness ...
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Continuity of evaluation maps in the topology of compact convergence on $C([0,\infty),\mathbb{R}^{n})$

I'm trying to prove that the evaluation maps $e_{x}:C([0,\infty),\mathbb{R}^{n})\rightarrow\mathbb{R}^{n}$ given by $e_{x}(f):=f(x)$ are Lipschitz-continuous with respect to the metric $\left|f-\...
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concepts which is present in metric space but not in topological space

I want to know some concepts which is present in metric space but not in topological space. The one that first comes to mind is uniform continuity, equicontinuity i.e. concepts defined with some kind ...
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A problem similar to Banach fixed point theorem

a) Let $(X,d)$ be a complete metric space and let $T: X \to X$. Prove that if there exists a natural $n$ such that $T^n(x)$ (composition of $T$ $n$ times) is a contraction then $T(x)$ has a unique ...
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Prove ${x:d(x,p) < d(x,q)}$ is open in metric space $X$

$X$ is a metric space and $p \neq q$ $\in X$. I want to prove that $E=$ $\{x:d(x,p) < d(x,q) \}$ is open in metric space $X$. I think I can directly prove this by showing every point $x \in E$ ...
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Metric Spaces: closure of a set is the set of all limits of sequences in that set

I am studying metric spaces and got confused about many different ways of defining the closure. Let $S$ be a subset of $M.$ Then, the closure of $S$ is $ \{x \in M : \forall \epsilon>0, \ \ B(x,\...
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Every metric space is a D-space.

I think it is correct, but I would like another pair of eyes to verify. Definition. An open neighborhood assignment is a function $f:X\to \tau$ such that $x\in f(x)$. Definition. A space is said to ...
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Continuity set of a difference of two upper semi-continuous real functions over a metric space

The difference of two upper semi-continuous functions is in general neither upper- nor lower semi-continuous. But what can be said about the continuity set of such a functions, specifically its ...
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non-separable metric space and measurablility of its elements

I'm studying Skorokhod space, which consists of cadlag functions, and I encountered the following sentence: If a metric space $(\mathbb{S}, \mathcal{S}, d)$ is not separable, then functions that ...
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If $(X,d')$ is totally bounded and $d'$ and $d$ are topologically equivalent then $(X, d)$ is separable

I am trying to write something similar to the proof of If $(X,d)$ totally bounded then $(X,d)$ separable but I dont know how to use topological equivalence here. Any help?
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A metric space is complete if for some $\epsilon \gt 0$, every $\epsilon$-ball in $X$ has compact closure.

This is a problem from Munkres' Topology. Let $X$ be a metric space. (a) Suppose that for some $\epsilon \gt 0$, every $\epsilon$-ball in $X$ has compact closure. Show that $X$ is complete. (b) ...
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Why are every structures I study based on Real number?

I've been studying basic concepts of inner product vector space, normed vector space and metric space. And all the inner products, norms and metrics are defined to be real-valued functions in my ...
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Are pseudocompact metric spaces complete?

Is there a way to show that pseudocompactness on a metric space implies completeness directly (without using sequential compactness)?
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Cauchy sequence in metric space

Give an example of a metric space such that a Cauchy sequence in $M$ that is not convergent. How can we give a example of that?
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Isometries of Metric Spaces

For a metric space $(X,d)$, let $\def\Iso{\operatorname{Iso}}\Iso(X,d)$ denote the group of bijective isometries of $(X,d)$. Clearly, $\Iso(X,d)$ is a group under composition. Question: Let $X$ be a ...
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Nonobvious examples of metric spaces that do not work like $\mathbb{R}^n$

This week, I come to the end of the first year analysis, and suffer from a "crisis of motivation." With this question, I want to chase away my thought, "Why is it important to study the general ...
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If (X,d) is a separable metric space then there exists a metric d′ that is topologically equivalent to d and such that (X,d′) is totally bounded.

I know that this question Separability, total boundness and topological equivalence of metrics has been asked, but the only solution given is not valid. There is something I already knew: (Y, d2) ...
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A typical example of Homeomorphism

The set $\mathbb{R}^2-\{(0,0)\}$ with the usual topology is: (A) Homeomorphic to the open unit disc in $\mathbb{R}^2$ (B) the cylinder $\{(x,y,z)\in \mathbb{R}^3/ x^2+y^2=1 \}$ (C) the ...
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A book with heuristics or general techniques used in real analysis?

I have been looking for a book with some good heuristics for real analysis and point set topology. Any ideas?
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Show that the sequence $(A_n)_{n≥1}$ in $L(l_1)$ does not converge to zero

For any $n ≥ 1$, define a linear operator $A_n : l_1 → l_1$ by $$A_nx = (0, . . . , 0, x_{n+1}, x_{n+2}, . . .), ∀x = (x_1, x_2, . . .) ∈ l_1.$$ Show that For any $x ∈ l_1$, we have $\lim_{n→∞} A_nx ...
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Open or closed status of addition of two subsets of a metric space

Question Let A and B be subsets of $R^n$. Define A + B = {a + b | a ∈ A, b ∈ B}. Consider the following sets W = {(x, y) ∈ $R^2$| x > 0, y > 0}, X = {(x, y) ∈ $R^2$ | x ∈ R, y = 0}, Y = {(x, y) ∈...
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Separability, total boundness and topological equivalence of metrics

The problem is: If $(X,d)$ is a separable metric space then there exists a metric $d'$ that is topologically equivalent to $d$ and such that $(X,d')$ is totally bounded. I know that if $(X,d)...
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Are points in different connected components separated by open subsets?

Decide if the following statement is true or false: If $a,b \in M$ belong to different connected components, then there exists a disconnection $M = A \cup B$ (with $A$, $B$ open and disjoint), ...
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Physical Meaning of Minkowski Distance when p > 2

Suppose we have two vectors in $u, v \in \mathbb{R}^d$. For $p \geq 1$, the Minkowski distance between these vectors is defined as $ \lVert u - v \rVert_p = \Bigl( \sum_{i=1}^d \lvert u_i - v_i \...
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Diameter of union is at most the sum of diameters, when the sets intersect

Prove that $d (A \cup B) \leq d(A)+ d(B)$, given that $ A \cap B \neq \varnothing$. Here $d$ stands for the diameter of the set. Please note that my knowledge is limited to metric spaces only, ...
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Question about notation of sequences and equivalence classes.

In these notes (see pg3 second-to-last paragraph), what does $d(x_k,x^\ast_{N_k})$ mean? The term $x_k$ lies in $X$, but $x^\ast_{N_k}$ is a class of Cauchy sequences in $X$. Should I take $d(x_k,x_{...
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Definition of onto for linear transformation

I had a question ask the following: "A linear transformation is onto if and only if the columns of its standard matrix form a generating set for its range." To me that seems true but the answer was ...
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Consider $X=C[0,1]$ with its usual sup-norm topology.Let $S=\{f\in X :\int _0^1f\neq 0\}$.Is the set connected?

Consider $X=C[0,1]$ with its usual sup-norm topology.Let $S=\{f\in X :\int _0^1f\neq 0\}$.Is the set connected? I tried to conclude from the path connectedness of $S$ .But $S$ is not path connected
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A continuous integer-valued function on a compact metric space has finite range

Let $X$ be a compact metric space and let $f:X\to\mathbb Z$ be a continuous function. (Here $\mathbb Z$ has the Euclidean topology induced from $\mathbb R$.) Prove that $f$ can assume only finitely ...
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A metric and discrete topology

Let $\Sigma=\{1,2,...,n\}$ and $\Omega=\Sigma^\mathbb{N}$ be the set of infinite sequence of n digits. Define a metric $d$ on $\Omega$ by $d(\omega,\tau)=2^{-|\omega\wedge\tau|}$ where $|\omega\wedge\...
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Is there a metric in which 1+2+3+4+… converges to -1/12?

It is well known that the sum $1+2+3+4+\ldots$, which tends to infinity in the regular sense, can be assigned the value $-\frac{1}{12}$ by different means, e.g., zeta regularization or different ...
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Erwin Kreyszig's Introductory Functional Analysis With Applications, Section 2.5, Problem 10

Here's Problem 10 in Section 2.5 in Introductory Functional Analysis With Applications by Erwin Kreyszig: Let $X$ and $Y$ be metric spaces, let $X$ be (sequentially) compact, and let the mapping $...
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Induced topology by a complete uniform space.

I know that Uniform space is generalization idea of metric space,Uniform space like metric space induce a topological space. Now my question is ( or are ):- In case our Uniform space was complete ,...
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Convergence of a series of vectors in a Banach space

Let $\sum_{k=1}^\infty\lambda^{k-1}\boldsymbol{v}_k$ be a series of vectors where $\lambda^{k-1}\in\mathbb{C}$, or $\lambda^{k-1}\in\mathbb{R}$, and the $\boldsymbol{v}_k$ belong to a Banach space. I ...
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Can a non-discrete metric $d$ be always defined on any non-empty set $X$ such that $(X,d)$ becomes a complete metric space? [closed]

Can a non-discrete metric $d$ be always defined on any non-empty set $X$ such that $(X,d)$ becomes a complete metric space ?
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Characterization of compact subsets in the metric space of all complex-valued sequences

Here's the statement of the Problem 4 after Section 2.5 in Introductory Functional Analysis With Applications by Erwine Kryszeg: Show that for an infinite subset $M$ in the space $s$ to be ...
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Metrizable compact spaces and Hausdorff spaces with a countable network

I have two questions related to metrizable spaces and countable network ; Can we find a continuous mapping from a separable metric space onto a non metrizable compact Hausdorff space. If a Hausdorff ...
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Reiterate Volterra integral operator is a contraction

I read in Kolmogorov and Fomin's Элементы теории функций и функционального анализа (p. 472 here) the statement that Volterra operator $A:L_2[a,b]\to L_2[a,b]$ defined by$$(A\varphi)(s):=\int_{[a,s]}K(...
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Is the cardinality of the set of all isolated points in a second countable metric space at-most $\aleph_0$?

Is the cardinality of the set of all isolated points in a second countable metric space at-most $\aleph_0$ ?