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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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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1answer
68 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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0answers
14 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
50 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 ...
4
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1answer
192 views

Completeness of the space of sets with distance defined by the measure of symmetric difference

Let $m$ be the measure defined on the set semiring $\mathfrak{S}_m$ and $m'$ its extension to the minimal ring $\mathfrak{R}(\mathfrak{S}_m)$. I read that $m'(A\triangle B)$ can be used as a distance ...
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0answers
44 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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2answers
21 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
30 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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0answers
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 ...
2
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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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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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2answers
92 views

TVS: Topology vs. Scalar Product

I just had an idea: It is clear that every scalar product induces a norm and that a metric and that finally a topology. Turning this argument around we know: Not every topology induces a metric only ...
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0answers
25 views

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
44 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
32 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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0answers
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
43 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 ...
3
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3answers
48 views

Problem 2.5.10 in Kreyszig's Functional Analysis Book

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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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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4answers
58 views

How to exhibit the set of all the limit points of this subset of $\mathbb{R}^k$?

Let $k$ be a positive integer, let $p_0$ be a point in $\mathbb{R}^k$, let $\delta_0$ be a positive real number, and let the set $E$ be defined as follows: $$E \colon= \{ \, p\in\mathbb{R}^k \, ...
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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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5answers
342 views

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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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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1answer
133 views

Ruler-and-Compass metric

Let $\nu \left( x \right) $ be the least number of steps that is required to construct a constructible length $x$, using compass and ruler in the well known fashion. Now, define the distance ...
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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 $x \in E$ ...
3
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1answer
29 views

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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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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460 views

Sequence has a convergent subsequence in R^n

Suppose A is a closed and bounded subset of R^n. Let {ak} be a sequence in A. Thus, the elements of {ak} are: (a11,a12,...,a1n), (a21,a22,...,a2n), ... ... (ak1,ak2,...,akn), ... We are not sure if ...
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2answers
36 views

Continuity set of a difference of two upper semi-continuous real functions over a metric space

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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1answer
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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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0answers
9 views

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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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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1answer
53 views

Irrational numbers in [0,1] [closed]

Why iirational numbers in interval [0,1] can't be countable union of closeds?
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2answers
82 views

Completing a metric space

This is captured from a chapter talking about completion of metric space in Real Analysis, Carothers, 1ed: Definition of completion: Actually, I cannot understand some parts of the proof. ...
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2answers
38 views

Let A and B be disjoint closed subsets of Rn. Define d(A,B)=inf{∥a−b∥:a∈A and b∈B}. Show that if A={a} is a singleton, then d(A,B)>0.

Let $A$ and $B$ be disjoint closed subsets of $\mathbb{R}^n$. Define $d(A,B)=\inf \{||a-b||: a \in A, b \in B\}$. I have to show that if $A=\{a\}$ is a singleton set, then $d(A,B)>0$ and I have no ...
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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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1answer
51 views

Are $d_{\infty}$ and $d_{p}$ distance functions?

Let $X$ be a set equipped with a metric $d_x$, denoted by $\langle X,d_x\rangle$, and $Y$ equipped with a metric $d_y$, denoted by $\langle X,d_y\rangle$. Let $Z=X\times Y$. Let ...
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1answer
26 views

Finding open balls in $\mathbb{R}^2$

If anyone can help I would be highly grateful! The Problem is in the image below.. [1] http://i.stack.imgur.com/eKDH2.png Should you approach using the open balls to find the boundary of the set? ...
3
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2answers
248 views

fixed point of homeomorphism and compactness of a complete metric space.

I need to know that the following statements if true or false: Every homeomorphism of $S^2\rightarrow S^2$ has a fixed point. Let $X$ be a complete metric space such that distance between any two ...
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2answers
31 views

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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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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Show that two metric space are equaly [closed]

Let $X$ be metric space with metric $d$ and $A \subseteq X$, $A$ is compact and not empty. Then for every non empty and closed set $B$, there exists a point $P\in A$, such that $d:(p,B)=d(A,B)$.
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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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311 views

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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4answers
42 views

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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1answer
39 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
40 views

Continuity of unique solution to differential equation

Let $f$ be a continuous function on $G$, where $G \subseteq \mathbb{R}^2$ is an open set containing $I \times [a,b]$ where $I:=[x_0-d,x_0+d]$, for some $a,b,d \in \mathbb{R}$ s.t. $a<b$ and ...
2
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1answer
21 views

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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60 views

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 ...