Metric spaces are sets on which you can measure the "distance" between any two points. The distance measurement is generally required to be symmetric (so distance from $A$ to $B$ is the same as distance from $B$ to $A$), positive for two distinct points, and obeying the triangle inequality.

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How is a metric space a topological space? [duplicate]

I learned about metric spaces and topological spaces but I don't see how they correlate. How does a metric space follow the properties of a topological space.
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2answers
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Prove that a set in a metric space cannot be both open and closed.

If I have a metric space $X$, and $E \subset X, E \ne X, E \ne\emptyset$. I want to prove that E cannot be both open and closed. I have two strategies, but I am not able to finish them: I assume ...
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134 views

Absolute convergence in a metric space

Let $(X,d)$ be a metric space, $(a_n)$ and $(b_n)$ are sequences in $(X,d)$. If $\sum_{n=1}^\infty d(a_n,b_n)$ is absolutely convergent, what do I say about the convergence of $(a_n)$ and $(b_n)$?
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3answers
36 views

Metrics (Distances) on $\mathbb{F}$ Theorem Proof

I had a question regarding a Theorem I had come across that described metrics (distances) on ordered field $\mathbb{F}.$ Here it is: Theorem: If $\mathbb{F}$ is an ordered field, then $d(x,y)=|x-y|$ ...
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77 views

Is $d(x,y)=|x-y|^2$ a distance on $\mathbb{R}$?

Please how to prove that $d(x,y)=|x-y|^2$ is a distance on $\mathbb{R}$, I don't know how to solve the triangular inequality. Thank you.
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1answer
30 views

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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3answers
53 views

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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37 views

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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1answer
34 views

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 ...
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49 views

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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1answer
31 views

Euclidean metric formula

Is this the correct formula for the euclidean metric (in $R^4$ )? $g_E = dr^2 + r^2(d\theta^2 + d\phi^2 + d\tau^2 + \cos \theta d\tau d\phi).$ I have been doing some calculations that are wrong and ...
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32 views

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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0answers
64 views

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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Show existence of a sub-sequence $(f_{n_k})$ which is uniformly convergent to a function in $C[0,1]$

Let $f_n:[0,1]\rightarrow R$ be a sequence of continuously differentiable function, Let $M>0$ be such that for any $0\le x \le 1$ and natural $n$, $|f_n(x)|$, $|f'_n(x)|<M$ Show existence of a ...
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1answer
34 views

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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1answer
47 views

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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1answer
36 views

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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32 views

Product spaces $X = Y = \mathbb R$

Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces. It is defined for $d_{X \times Y} : X \times Y \rightarrow \mathbb R_+$ with $$d_{X \times Y}((x_1,y_1),(x_2,y_2)) := d_X(x_1, x_2) + d_Y(y_1, y_2)$$ ...
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1answer
44 views

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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1answer
23 views

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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1answer
16 views

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 ...
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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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2answers
22 views

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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2answers
43 views

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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1answer
23 views

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, \ \ ...
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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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2answers
37 views

Continuity set of a difference of two upper semi-continuous real functions over a metric space [on hold]

I wanted to know if we can get some properties of the continuity set of a difference of two upper semi-continuous real functions over a metric space? Or maybe for a restriction?.
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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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Does nonexpansive mapping imply isometry in this case?

I have the following problem. I want to prove that there exists an isometric isomorphism: $$Lip_0(X) \equiv AE(X)^*$$ Here $(X, d)$ is a metric space, $Lip_0(X)$ is the space (a Banach space with the ...
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1answer
27 views

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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10answers
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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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1answer
40 views

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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1answer
17 views

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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1answer
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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 ...
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25 views

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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2answers
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Prove that $d (A \cup B) \leq d(A)+ d(B)$

Before I begin, please note that my knowledge is limited to metric spaces only, with no knowledge of topology at all. Now, my attempt: I'm trying to use the relation that $A \subset B \Rightarrow ...
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1answer
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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 ...
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1answer
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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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Why are closed discs preserved by linear fractional transformations in non-archimedean geometry?

If $K$ is a local field equipped with a non-archimedean metric, then I have read in several places that the action of $PGL_2(K)$ by linear fractional transformations takes closed discs to closed ...
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2answers
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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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1answer
29 views

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 ...