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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Question on pointwise convergence of Cauchy sequences

Let $B = (B(X, \mathbb{R}), d_u)$ be the set of all bounded functions from the metric space $X$ into $\mathbb{R}$. Let $(f_n(x))$ be a Cauchy sequence in $B$. Is the following statement valid - As ...
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Proof that a subspace $A$ of a complete metric space $X$ is complete iff $A$ is closed

Here's my proof in my own words, does it stack up? Showing $A$ is complete implies $A$ is closed. Let $(x_n)$ be a convergent sequence in $A$. $A$ is complete $\implies (x_n) \to p \in A$. Hence $A$ ...
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Find the diameter of a set faster than $O(|M|^2)$

$M$ is a finite set of points in a metric space. I want to calculate the diameter of the set, i.e. the greatest distance between two points. Is there a smarter way to do this than to calculate the ...
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2answers
239 views

What does it mean to be a “closed subset of a metric space”?

So I am working my way through the Dover book, Intro to Topology by Bert Mendelson, and in the section on open and closed sets, I'm stuck on the following notation for this problem: Let $(X,d_1)$, ...
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Metric on space of functions

For a set $X$, let $\mathbb{R}^{X}$ be the set of all maps from $X$ to $\mathbb{R}$. For $f,g\in\mathbb{\mathbb{R}}^{X}$, define $$d(f,g) = \sup_{x\in X}\frac{|f(x)-g(x)|}{1+|f(x)-g(x)|}.$$ I am ...
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How to prove that the set of condensation points of an uncountable subset of the real euclidean k-space is perfect?

I'm referring to Problem 27 in the exercises of Chapter 2 in the textbook, Principles of Mathematical Analysis, 3rd edition, by Walter Rudin. I've managed to prove that the set $P$ of condensation ...
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compact-open metrizability

Given topological spaces $X$ and $Y$ the set $C(X,Y)$ of all continuous functions $f:X\to Y$ becomes a topological space with the compact-open topology (that is the topology generated by the sets ...
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Completeness for product topolgy

Let $(X_n, d_n)$ be a countable family of metric spaces. Show that the distance function on the product topology defined by $$d\big( (x),(y) \big) = \sum_{n\geqslant 1} 2^{-n} \frac{d_n(x_n,y_n)} ...
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What is straight line?

I have found the definition of line in metric space. It is general but has two problems. Considering about $\mathbb R^2$ equipped rectilinear distance, every line by this definition contains a ...
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Showing a differential equation has a unique solution in $C[0, 1]$

Show that $$F(f)(t) = t^2 + \frac{t}{3}f(t) + \frac{1}{5}\int_0^t e^uf(u) du$$ is a contraction on $(C[0, 1), d_u)$. Deduce that the differential equation $$(15 − 5t)\frac{df}{dt} = (5 + 3e^{t})f + ...
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Complete metric space - sequences

The problem I'm trying is to prove whether or not the metric space of real-valued sequences $(x_n)$ such that $x_n=0$ for all but finitely many values of $n$, with the sup metric: $d((x_n),(y_n)) = ...
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2answers
291 views

Limit of sequences in metric space

I could not prove the following statement. Can you help me? Let $X, d(x, y)$ be a metric space, and let $(x_n)$ be a sequence of points in $X$. Prove that $x_n → a$ if and only if for every open set ...
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1answer
333 views

Boundary and interior of a set, proof help (metric space)

Could anybody give me a hand to prove the following question that I have just seen on the book? I really appreciate your help! Let $X$, $d(x, y)$ be a metric space and $A ⊂ X$, $B ⊂ X$. Prove the ...
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Distance of two sets and their closest points

Assume we have a metric space $S$, a metric $d$ and two subsets of it, called $A$ and $B$. Assume also that $A\cap B = \emptyset$. Assume also that at least one of these sets is bounded, ie. has no ...
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Characterisation of Cantor-connectedness

For Cantor-connectedness I use the following definition: A $p$-metric space $(X,d)$ is Cantor-connected if for any $\epsilon > 0$, any two points $x, y \in X$ can be connected by an ...
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Two metrics have the same open sets (proof help needed)

I could not find a way to prove the following. Could you please help me? Regards Let $X$ be a set, and let $d_1(x, y)$ and $d_2(x, y)$ be two metrics in $X$. Suppose that the metrics $d_1(x, y)$ and ...
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Should every line be infinite in both two directions?

Yesterday one my classmate asked me for the the definition of line. I have found it in coordinate geometry, in vector space, and in metric space. The definition in metric space is quite general ...
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A question about a continuous function on a compact metric space

suppose that (X,d) is a compact metric space and $f:X\to X$ is a continuous function. Define $X_1 = f(X), X_2 = f(X_1),...,X_{i+1} = f(X_i),...$ and let $A = \bigcap_{i=1}^\infty X_i$. Is $A \subseteq ...
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2answers
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Lipschitz property and Lipschitz extension

Is there a Lipschitz function $f$ from a subset of a metric space $U$ to a complete metric space $V$ that has no Lipschitz extension to the whole space $U$?
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pseudo-inverse to the operation of turning a metric space into a topological space

The construction of turning a metric space $(X,d)$ into a topological space by inducing the topology generated by the open balls gives rise to a functor $Met\to Top$ for any reasonable category $Met$ ...
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1answer
94 views

non-Compact metrizable implies a countable closed discrete subspace?

I'm working through a proof in which $(X,\tau)$ is a $T_1$-space that is metrizable. The author says, "since $(X,\tau)$ is metrizable, there is a countably infinite, closed (in $X$) and discrete ...
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Showing the distance between sets is indeed a metric.

Let $(X,d)$ be a metric space, and let $A$ be a non-empty subset of $X$. Define the distance between a point and a set by the function $$d(x,A) = \inf_{z \in A}d(x,z).$$ Prove that for all $x,y \in ...
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251 views

Continuity of metric space of integrals of continuous functions

Let $R$ be the real line with the standard metric $d:R \times R \to R$ be defined by $d(x,y) = |x-y|$. Let $X$ be the set of continuous functions $f:[a,b] \to R$ of an arbitrary closed interval ...
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1answer
80 views

How to prove the space of bounded linear functionals is complete?

How to prove the space of bounded linear functionals in a linear norm space is complete? I just have no idea how to use assumptions about $f_n$ to prove $f$ is bounded? I do it like this, Main idea ...
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1answer
202 views

Covers, and compact sets

Please go to this lecture(1). I have numbered my questions for organization and added the picture which I will constantly refer to. I also apologize in advance for this lengthy question Right about ...
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97 views

Decreasing sequence of closed set in a metric space is convergent?

Let $\{E_n\}$ be a collection of bounded and closed subsets in a metric space $X$ such that $E_{n+1} \subset E_n$ and $lim_{n\to\infty} diam E_n = 0$. It's a theorem that if $X$ is complete, then ...
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202 views

Connected length space with disjoint open ball property

This question deals with a special case of this question, which has not yet been satisfactorily solved. If you have any ideas about that general case, feel free to answer there and I'll be happy to ...
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2answers
240 views

Connected sets.

Fix any number $\delta>0$ and put $A = \{x \in \mathbb{R}: \left|x-3\right|<\delta\}$ and $B = \{x \in \mathbb{R}: \left|x-3\right|>\delta \}$. Prove that $C=A \cup B$ is not a connected ...
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Prove if one set is complete then another set is complete

Let $X$ be a set. Let $l^{\infty}(X,N)$ be all bounded functions on the form $f: X\longrightarrow N$. Let $d(f,g)=\sup\{n(f(x),g(x): x\in X)\}$ be a metric on $l^{\infty}(X,N)$, where $n$ is metric on ...
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68 views

Show that $C([0,1],\mathbb{R})$ with the $L_2$ inner product norm is not a Hilbert space.

I need to prove that all continuous functions on the closed set $[0,1]$ is not a Hilbert space. Given the $L_2$ norm. I guess I need to show that every Cauchy sequence in the space, does not ...
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1answer
570 views

Show that a connected metric space is $\epsilon$-chainable for $\epsilon>0$

Show a connected metric space (X,d) is $\epsilon$-chainable for $\epsilon >0$ $\epsilon$-chainable Definition (X,d) is $\epsilon$-chainable if given any two points $a,b \in X$, $\exists$ ...
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$A = \{(x,y) \in \mathbb{R^2}: y = \frac{1}{x}, x > 0\}$. Show $A$ is closed in $\mathbb{R^2}$

Given $A = \{(x,y) \in \mathbb{R^2}: y = \frac{1}{x}, x > 0\}$. Show (by considering convergent sequences or otherwise) $A$ is closed in $\mathbb{R^2}$. Anyone able to give me some advice on how ...
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3answers
77 views

Showing $f:\mathbb{R^2} \to \mathbb{R}$, $f(x, y) = x$ is continuous

Let $(x_n)$ be a sequence in $\mathbb{R^2}$ and $c \in \mathbb{R^2}$. To show $f$ is continuous we want to show if $(x_n) \to c$, $f(x) \to f(c)$. As $(x_n) \to c$ we can take $B_\epsilon(c)$, ...
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Showing equivalent metrics have the same convergent sequences

I am trying to show two equivalent metrics $p$ and $d$ on a set $X$ have the same convergent sequences. $p$ and $d$ are such that $kd(x,y) \leq p(x,y) \leq td(x,y)$ for every $x, y \in X$, $k$ and $t$ ...
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$L_p$ complete for $p<1$

It is rather straight forward to show that $L_p$ is complete for $p\geqslant 1$, but I am having trouble showing the same thing when $p<1$. For the former case I have shown that every absolutely ...
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shows that $d$ is continuous where $d$ is a metric defined on $X$

If $(X; d)$ is a metric space, then the metric $d$ on $X$ induces a product metric $p$ on $X\times X$ by $p((x_1; y_1); (x_2; y_2)) := d(x_1; x_2) + d(y_1; y_2)$ Show that $d :(X \times X ) ...
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67 views

Coinciding with the Product Topology

I am a bit confused by this whole question I have in front of me. It defines a distance, $d$, on a product topology $X= \Pi_i X_i $, where $\Pi_iU_i$ forms a basis of open sets and $U_i=X_i$ except ...
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3answers
387 views

Meaning of closure of a set

Does closure of a set mean, only adding boundary values if the set is open and leave it as it is if the set is closed?
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Volume and diagonal length of the Hilbert cube

Here's something sort of fun that I gave thought to a while ago, and now that I've done some maturing mathematically I'm curious to see if my musings are legitimate. Let $H=[0,1] \times ...
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Cantor set - a question about being metrizable and about the connected components

I have a question regarding Cantor set given to me as a homework question (well, part of it): a. Prove that the only connected components of Cantor set are the singletons $\{x\}$ where $x\in C$ ...
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$C$-doubling $\Bbb R^2$ measure gives measure zero to a straight line?

A metric space $X$ with metric $d$ is said to be doubling on $\Bbb R^2$ if there is some constant $C > 0$ such that for any $x \in X$ and $r > 0$, the Euclidean ball $B(x, r) = \{y:|x − y| < ...
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Is this the category of metric spaces and continuous functions?

Suppose the object of the category are metric spaces and for $\left(A,d_A\right)$ and $\left(B,d_B\right)$ metric spaces over sets A and B, a morphisms of two metric space is given by a function ...
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Coinciding open sets

I'm given two distances that are defined on some metric space and I need to show that open sets and Cauchy sequences coincide for the two distances. What does this mean? I'm avoiding giving details on ...
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Hausdorff dimension

Could you please give me some hints on (Exercise 1.7.21) of "A Course in Metric Geometry" by Burago, Burago, Ivanov. We have a compact space $X$, which can be written as $X=\bigcup_{i=1}^n X_i$ ...
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Is {$\phi$} set forms a metric space or not?

Is $\phi$ set forms a metric space or not ? I think, it does not form a metric space, because, we can't specify a metric on $\phi$. But, In many text book, it is not mention that, the set on which, ...
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At most countable subsets of a compact metric space.

As written, the question is: Let (X,d) be a compact metric space. Prove that for each $\epsilon>0$ there exists a positive integer $N$ such that for each $S \subseteq X$, if $S\thicksim Z_N$, then ...
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3answers
323 views

Relationship between sequences and closed sets

I seem to recall that you can say a set is closed if there exists a sequence that converges to a limit point of that set...obviously that is not correct but the idea is that you can deduce a set is ...
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Prove that a compact metric space can be covered by open balls that don't overlap too much.

The problem is: For compact metric space $(X,d)$ prove that for every $r>0$ there exists a subset $S$ of $X$ such that $\{\mbox{Open balls of radius }r\mbox{ centered at }p \mid\mbox{ for all }p ...
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1answer
786 views

Closed subset of complete metric space…don't understand last part of theorem.

A closed subset of a complete metric space is a complete subspace. Proof. Let $S$ be a closed subspace of a complete metric space X. Let $(x_n)$ be a Cauchy sequence in $S$. Then $(x_n)$ is a ...
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1answer
67 views

Show that $[x, y]$ is complete where $x < y$ in $\mathbb{R}$

As the interval is closed every sequence in the interval converges to some point $x$ in the interval, and every convergent sequence is a Cauchy sequence, hence $[x, y]$ is complete. Is that correct? ...