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 this topological space different from the euclidean one?

I'm preparing for my topology exam and came across this example which I can't figure out. Let $\mathcal{T}$ be a such family of all sets $U\subset \mathbb{R}^2$ that $U\cap L$ is an open set in L ...
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16 views

Continuity of functionals in function space

I came across this problem and got confused. With the help of folks at MathStackExchange I managed to understand the following: Define $h:C[0,1]\rightarrow \mathbb{R_+}$ by $$h(x)=\sup_{0\leq ...
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continuity of functional in $C[0,1]$

I came across this problem and got confused Problem: Define $h:C[0,1]\rightarrow \mathbb{R_+}$ by $$h(x)=\sup_{0\leq t\leq1}|x(t)|$$Show $h$ is continuous in $C[0,1]$ Attempt: I am a bit confused ...
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31 views

Show that $\lbrace 1-\frac{1}{n} \rbrace_n$ does not converge in the Sorgenfry topology.

Consider $\{1-\frac{1}{n}\}_n=\{0,\frac{1}{2},\frac{2}{3},\frac{3}{4},\cdots\}$. If $\{1-\frac{1}{n}\}_n$ converges, then $\{1-\frac{1}{n}\}_n \rightarrow 1$. If it converges, then, by definition, ...
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20 views

Rational linear approximation in $C[0,1]$

I came across the following question in my homework and have been stuck for a long time Problem: Let $C_0$ be the countable subset of $C[0,1]$ consisting of all piecewise linear functions with ...
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41 views

A question about compact sets: how to prove $g$ must be an isometry

Let $(X,p)$ be a compact metric space. Suppose that $g:X\rightarrow X$ is a function such that for all $x_1,x_2\in X$ we have $p(g(x_1),g(x_2))\geq p(x_1,x_2)$. Prove that, in fact, $g$ must be an ...
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7 views

Conditions on the metric function in a flat manifold

It is well known that a manifold is flat iff its Riemann tensor vanishes identically. However, the equation $R^{\mu}_{\nu\rho\sigma}=0$ is a differential equation for the metric tensor $g_{\mu \nu}$. ...
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14 views

Existence of CAT(0)-metrics

Given a metrisable, contractible topological space. Under which conditions exists a CAT(0)-metric compatible with the given topology?
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25 views

Metric space problem.

Suppose $S$ is a subset of $\mathbb{R}$ and $z$ belongs to $\mathbb{R}$. Then prove that dist $(z, S) \leq |z-\sup S|$ with equality if $z\geq \sup S$. I can easily prove it for $S=\varnothing$. But ...
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23 views

Finding connected componets of a set of continuous functions

In the metric space (C[0,1], d∞) consider the set: U= {f in C[0,1]: f(x)≠0 for all x in [0,1]} Prove that U is open and find its connected components. Proving that U is open is easy, but I don't ...
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When talking about a normed vector space, does it's metric always need to be the induced one?

The title basically says it all. If we have a normed vector space, is it possible to work with the space as a metric space with a different metric than the induced one? So if the space is $(X,||\ ...
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Theorem 3.54 (about certain rearrangements of a conditionally convergent series) in Baby Rudin: A couple of questions about the proof

Here's the statement of Theorem 3.54 in Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Let $\sum a_n$ be a series of real numbers which converges, but not absolutely. Suppose ...
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15 views

Positive definite functions defined on the embedding of a planar graph in the plane

By way of motivation: Let $f:[0,1]\rightarrow\mathbb{R}^2$ and $g:\mathbb{S}_1\rightarrow\mathbb{R}^2$ be continuous injections, where $\mathbb{S}_1$ is the circle ($\mathbb{R}/\mathbb{Z}$). Then, ...
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29 views

The closed and bounded sets are compact in the product topology

Let $X=\mathbb{R}^{\aleph_0}$ with the product topology, it is true that all the closed and bounded (in the uniform sense) sets are compact?
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3answers
41 views

Show that the sequence of functions $(x_n)_{n≥1}$ in $C[0, 1]$ given by $x_n(t) = t^{2n} − t^{3n} , ∀t ∈ [0, 1]$ is bounded

That is $C[0,1]$ equipped with the supremum metric. I have proven, using derivatives, that each function $x_n$ has a local maximum and local minimum at $(2/3)^{1/n}$ and $0$ respectively. I know ...
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Problem 30 in the Exercises following Chapter 2 in Baby Rudin: How to immitate the proof of Theorem 2.43?

Here's problem 30, the very last one, in the Exercises after Chapter 2 in Walter Rudin's Principles of Mathematical Analysis, 3rd edition: Imitate the proof of Theorem 2.43 to obtain the following ...
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1answer
27 views

Contraction Mapping, Metric

Let $X$ = {all continuous functions $f$:[0,1] $\rightarrow$ [0,1]} and let $d$ be the metric on $X$ given by $d$($f$,$g$)= $sup_{t\in[0,1]}$ |$f$($t$)-$g$($t$)| for $f$,$g$ $\in$ $X$ Show that ...
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2answers
62 views

Is the subset $[0, \sqrt2] ∩\mathbb{Q} ⊂ \mathbb{Q}$ closed, bounded, compact?

Letting $\mathbb{Q}$ be equipped with the Euclidean metric. What I can work out is that it is bounded as its contained in the closed ball of radius ${\sqrt2}/{2}$ centred at ${\sqrt2}/{2} $. Its not ...
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48 views

Contraction-like mapping without fixed point?

If $(X,d)$ is a complete metric space and $\xi:\;X\to X$ satisfies: $$d(x,y)<n+1\Rightarrow d(\xi(x),\xi(y))<n$$ $$d(x,y)<1/n\Rightarrow d(\xi(x),\xi(y))<1/(n+1)$$ for all $n= 1,2,\dots$, ...
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1answer
36 views

Closedness and boundedness in metrizable topological spaces

This is a quick question that I have not managed to answer myself: let $X$ be a metrizable topological space, let $A\subset X$ be a closed, bounded subset. $f:X\to Y$ is a homeomorphism, must $f(A)$ ...
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1answer
32 views

Example of metric space completion

I'm looking for examples of noncomplete metric spaces and their completions. I know of some basic examples like completion of open intervals and rational numbers(both with the reals and p-adic ...
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26 views

Complete subspace of continuous function from compact subset [on hold]

Assume $K\in \mathbb{R}$ compact. How to prove that $C^0(K,\mathbb{R})$ is complete. Where $C^0(\mathbb{R},\mathbb{R})$ is the space of continuous f from $\mathbb{R}$.
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Structure of the space $X:=\{x \subset \mathbb{R}^3: |x| < \infty\}$, where $|x|$ is the cardinality of $x$

I am interested in the space $$ X:=\{x \subset \mathbb{R}^3: |x| < \infty\}, $$ where $|x|$ is the cardinality of the subset $x$. This is basically configuration space for a quantum system with a ...
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1answer
32 views

Topological structure of the Manifold valued functions

$M$ is a Riemannian manifold. What condition on $M$ for $\mathcal{C}_{[a,b]}(M)$ (the set of continuous functions of the real interval $I=[a,b]$ to $M$) to be a polish space ? For which topology ? Is ...
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63 views

Show the subset $A$ of $\mathbb{R}^n$ is compact

Show the subset $$A = \{(x_1, . . . , x_n) ∈ \mathbb{R}^n| −1 ≤ x_1 ≤ x_2 ≤ · · · ≤ x_n ≤ 1\} \subset \mathbb{R}^n $$ is compact, and show the function $$\left\{\begin{array}{}f : A → ...
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14 views

Extension theorem for locally Lipschitz functions

Let A be a subset of a metric space $(X,d)$ and f be a real valued locally lipschitz function on A. Does there exists a real valued locally lipschitz function on X which is an extension of f? or under ...
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45 views

Irrational numbers in [0,1] [on hold]

Why iirational numbers in interval [0,1] can't be countable union of closed sets?
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Adjoint of completely continuous operator is completely continuous

In the proof of the fact that the adjoint operator $A^\ast$ of a completely continuous linear operator $A:E\to E$ mapping a Banach space into itself is also completely continuous on $E^\ast$ endowed ...
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32 views

How to prove a map between two spaces of real sequences $f : l^1 \to l^2 $ is well-defined and continuous

the question is whether the following statement is ture or false, and justify it. Here is the statement The map $f : l^1\to l^2$ given by $f(x_0, x_1, x_2,...)= (x_0, x_1, x_2,...) $ is ...
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0answers
7 views

Conditions for positive definiteness for a class of matrices induced by a semimetric

Let $X$ be a set, and let $d:X\times X\rightarrow \mathbb{R}$ be a semimetric on that set (i.e. $\forall x,y\in X$, $d(x,y)=d(y,x)\ge 0$, and $d(x,y)=0$ iff $x=y$). I seek conditions on $X$ and $d$ ...
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18 views

not metrizable?

In Munkres, section 30, exercise 6 is this: Show that $R_{l}$ and $I^2_0$ are not metrizable. I guess $R_{l}$ is lower limit topology, and $I^2_0$ is an ordered square. and here, how to prove they ...
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1answer
16 views

How to find the interior of this set?

let $S=\{A\in M_n(\mathbb R):tr(A)=0\}$ The question is to check whether $S$ is Nowhere dense .I think the set is closed and hence the problem reduces to findind int(S).How to do that?
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$\{(x,y)\in \mathbb R^2:xy=1\}$

To check which pairs are Homeomorphic? A.$\{(x,y)\in \mathbb R^2:xy=0\}$ B.$\{(x,y)\in \mathbb R^2:xy=1\}$ C.$\{(x,y)\in \mathbb R^2:xy=0,x+y\geq0\}$ D.$\{(x,y)\in \mathbb R^2:xy=1,x+y\geq 0\}$ I ...
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1answer
22 views

How to check the compactness of these sets:

How to check the compactness of these sets: a.the unit sphere in $l_2$ the space of all square summable real sequences with its usual metric i.e.$d({x_i,y_i}) =(\sum_1^\infty|x_i-y_i|^2)^{1/2}$ ...
3
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1answer
50 views

Erwine Kryszeg's _Introductory Functional Analysis With Applications_: Section 2.3, Prob. 14

Here's problem 14 in the Problem Set immediately following Section 2.3 in the book, Introductory Functional Analysis With Applications by Erwine Kryszeg. Let $Y$ be a closed subspace of a normed ...
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Is the sequnce of functions $f_n$ convergent?

let $f_n$ be a sequence of functions in $C[0,1]$ and they are differentiable continuously in $(0,1)$. Also $|f_n(x)|\leq 1$ and $|f_n{'}(x)|\leq 1$ forall $x\in [0,1]$ and for each $n$. Since $[0,1]$ ...
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0answers
30 views

Metrizable topological space

Why Extended real numbers set with T ( T topology on R with infinity) , is metrizable ? And how can prove that d(x,y) genetares this topology (T) ??
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2answers
58 views

Does this strengthening of continuity have a characterization in terms of familiar concepts?

Definition 0. Whenever $X$ is a metric space, $A \subseteq X$ is a subset, and $r \in \mathbb{R}_{>0}$ is a positive real number, define that $$A \oplus r = \bigcup_{a \in A} B_r(a).$$ Definition ...
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1answer
81 views

Are all the finite dimensional vector spaces with a metric isometric to $\mathbb R^n$

Are all the finite dimensional vector spaces with a metric isometric to $\mathbb R^n$? My goal is to claim that in any finite dimensional vector space, equipped with a metric, a closed-bounded subset ...
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2answers
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Sequences in Metric Spaces [closed]

Find a metric space $(X,d)$ and a sequence $(x_n)$ in X that has no convergent subsequences but for which the infimum of the set $\{d(x_m,x_m)\mid m$ and $n$ are distinct natural numbers$\}$ is zero. ...
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1answer
13 views

Multiple choice question on continuous function on a unit ball

Pick out true: Let $B$ be the closed unit ball and $D$ be the open unit ball. a.Given a continuous function $g:B\rightarrow \mathbb R$ there always exists a continuous function $f:\mathbb ...
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2answers
38 views

Canonical compactification of a metric space

There are many constructions to produce a compact metric space from an arbitrary metric space (sometimes extra conditions are imposed). But is it possible to compactify a metric space M into M* such ...
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2answers
40 views

Continuity of vector space operations in a normed space

Here's problem 4 immediately following section 2.3 in Erwine Kryszeg's book, Introductory Functional Analysis With Applications: Show that in a normed space $X$, vector addition and scalar ...
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0answers
27 views

Prove this map is continuous

$(rcos(t),rsin(t))↦((1/r).cos(t),(1/r).sin(t)), 0≤t≤2pi $ first for $0<r<1$, then for $r>1$ My idea is to say $(rcos(t),rsin(t)) = r .(cos(t),sin(t))$ then the cos and sin map with an ...
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Prove $d(x,y)=\sup _{n} \left| \sum _{k=1}^{n}(x_k-y_k)\right |$ is a metric

Let $\gamma$ be the set of convergent series.$$\gamma = \{x=(x_k), x_k \in \mathbb{R} : \sum x_k <\infty\}$$ Prove that $(\gamma , d)$ is a metric space, with $$d(x,y)=\sup _{n} \left| \sum ...
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2answers
24 views

Check if the parabola (with an induced topology) $\{(x,y)\in\mathbb{R}^2 | y=x^2\}$ is connected or compact.

i think yes connected but not compact, as it cannot be represented as a disjoint union and there is no finite sub cover. I'm just not sure how to go about proving this i.e. what to actually write ...
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2answers
36 views

Is the map, $ f:(0,1)⊂ \mathbb{R}$ → $(1,∞)⊂ \mathbb{R}$ : $x ↦ 1/x $continuous?

I feel it is, but cannot prove why. Also is it bijective, and is its inverse continuous?
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1answer
28 views

Making an interval with point deleted complete

I am playing around with metric and topological spaces to get a better grasp of them, and I am wondering the following: is it possible to have a metric such that the set $[-1,0)\cup (0,1]$ is complete ...
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1answer
56 views

Show these sets are homeomorphic to eachother

1) {${(x, y) ∈ R^2 |0 < x^2 + y^2 < 1}$} 2) {${(x, y) ∈ R^2 | x^2 + y^2 > 1}$} I've considered mapping r to 1/r, from (0,1) to (1,∞)
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1answer
43 views

Theorem 2.3-2 in _Introductory Functional Analysis With Applications_ by Erwine Kryszeg

Here's the statement of Theorem 2.3-2 in the book mentioned above: Let $(X,||\cdot||)$ be a normed space. Then there is a Banach space $\hat{X}$ and an isometry $A \colon X \to W$, where $W = A(X)$, ...