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 should one think about results that depend on AC?

I just encountered this: "(Theorem of A. H. Stone) Every metric space is paracompact... Existing proofs of this require the axiom of choice... It has been shown that neither ZF theory nor ZF ...
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83 views

a proof for dense subsets in metric spaces

Let $(X,d)$ be a metric space and $A \subset X$. If $\partial A=X$ then prove that $A$ is dense in $X$. And i also need an example for the converse is not always true.
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299 views

Differentiability in metric spaces

I have a question in mind: Why can't we define differentiability in arbitrary metric spaces? Or can we define it really? Please discuss. I only have studied the notion of differentiability in ...
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174 views

On the definition of a geodesic in a metric space

I am interested in the definition of geodesics in metric spaces. A definition which seems reasonable to me is that a geodesic should locally be a distance minimizer. Wikipedia ...
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310 views

Continuity and sequential continuity

Prove that: > The function $f:(X,d)\rightarrow(Y,\rho)$ is continuous if and only if $f$ is sequentially continuous (that means $x_n\rightarrow x \Rightarrow f(x_n)\rightarrow f(x)$) Proof. ...
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1answer
82 views

Definition of functions on metric spaces.

In the post Definition of functions, it is stated in the accepted answer that one way to define a function is to define it as the triple $(f, X, Y)$ where $f \subset X \times Y$. My question is what ...
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79 views

Variations in math to implement three-dimensional space?

Backstory: So I was researching topics, and found that 3-D game programming often markets itself with linear algebra. As a philosopher of math I decided to dig further into this and determine if ...
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1answer
83 views

Why does the natural quotient metric works in this case?

There have been at least two discussions in this forum about why a (pseudo) metric is defined in the quotient of a metric space by: $$ d([x],[y]) = \inf \{ d(p_{1}, q_{1}) + \ldots, d(p_{n}, q_{n}) \} ...
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1answer
47 views

Constructing a metric subspace of $\prod_{\alpha\in I}X_\alpha$, where $X_\alpha$ is a metric space

Suppose that for each $\alpha\in I$, $(d_\alpha,X_\alpha)$ is a metric space. (The indexing set $I$ is arbitrary.) I want to find a maximal subset $Y$ of the product space $X=\prod_{\alpha\in ...
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167 views

Continuous functions on metric spaces

I am considering a problem (from Goldberg, "Methods of Real Analysis") where f is a continuous injective map from an arbitrary metric space, (M,d), to the discrete metric space (the reals with the ...
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143 views

Is a Baire Space necessarily complete?

A complete metric space a Baire space. But is a Baire space necessarily complete?
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104 views

Continuous extension of $f$ from $E$ to $\mathbb{R}$

If f is a real continuous function defined on a closed set $E \subset \mathbb{R}$ , prove that there exist continuous real function g on $\mathbb{R}$ such that $g(x)=f(x)$ for all $x\in E$.
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Determine whether this d is a metric?

Suppose we have a function $d:\mathbb{Z}^{2}\times\mathbb{Z}^{2}\rightarrow\mathbb{R}$ where $\forall(x_{0},y_{0}),(x_{1},y_{1})\in \mathbb{Z}^{2}$, we have.. $d\left( \left( x_{0},y_{0} \right), ...
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55 views

Analogue of closed graph theorem

This is the analogue of closed graph theorem for compact space Suppose that $X$ and $K$ are metric spaces, that $K$ is compact, and that the graph of $f: X \rightarrow K$ is a closed subset ...
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3answers
65 views

To check $(0,1)$ is open in $(0,1] $ or not

We know $(0,1)$ is open in $\mathbb{R}$. Please explain if $(0,1)$ is open in $(0,1]$ or not. How to do that?
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1answer
56 views

definiton of open sets in metric spaces

For metric spaces, the definition of an open set $U\subset X$ is that it is a set which for any point $u\in U$ in the set there exists some $\epsilon>0$ such that the open ball ...
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154 views

Construct a complete metric on $(0,1)$

Can anyone construct a complete metric on $(0,1)$ which induces the usual subspace topology on $(0,1)$ ?
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1answer
137 views

Does every sphere defined by jungle/paris metric is a retract of $\mathbb{R}^2$ with jungle/paris topology?

In this case, firstly I'm being told that in ($\mathbb{R}^n$, $d_e$), where $d_e$ is the euclidean metric, every closed sphere defined by this metric $D^n(x_0, x)=\{x\in \mathbb{R}^n:\ d(x_0, r) \le ...
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1answer
56 views

Can a function be both upper and lower quasi-continuous?

Can you give me a non-trivial example? Below is the definition I am using: A function $f: X \rightarrow \mathbb{R}$ is said to be upper (lower) quasi-continuous at $x \in X$ if for each $\epsilon ...
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0answers
23 views

Extension of a metric space [duplicate]

Let Y be a subset of X. let (Y,d) be a metric space. Can we define a metric d* on X such that (Y,d) becomes a subspace of (X,d*)? If so, is the extension unique?
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47 views

Norms inducing non discrete Hausdorff topology

We know that any norm defined on a vector space V induces a non discrete Hausdorff Topology on V. Is the converse true?
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1answer
82 views

Proof of Metric spaces

Suppose that $p_1$ and $p_2$ are metrics on $M$. Prove that $p=\max\{p_1,p_2\}$ is a metric on $M$. I am supposed to define $p_2$ and $p_3$ as: \begin{gather*}p_1=\sup(x,y),\\ ...
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57 views

Finite range operator is compact

This theorem is from Rudin book which he says that obvious, but I'm quite confused how to prove it completely. Hope someone can help me clarify. Let $X$, $Y$ be Banach spaces, If $T \in ...
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1answer
75 views

Continuous function and nested compact spaces

Let $X,Y$ be metric spaces and $f:X \to Y$ be a continuous function. Let $K_n \subset X$ be a compact subspace of $X$ for $n \in \mathbb N$ such that $K_{n+1} \subset K_n$. Prove that $f(\bigcap_{n ...
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1answer
47 views

Detail on a theorem of continuity and compactness

I have come across a proof in a book. I have trouble convincing myself on a statement on said proof. The theorem is a well-known one. I am stating the version I found on Ross's "Elementary Analysis." ...
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76 views

idea for the completion of a metric space

While doing the proof of the existence of completion of a metric space, usually books give an idea that the missing limit points are added into the space for obtaining the completion. But I do not ...
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1answer
25 views

Two ways to express boundedness

I'm little confused about the boundedness in a Banach space. Here are two boundedness definition we can encounter in Banach space: 1) A set $E$ is bounded if, for every neighborhood of 0, we have $E ...
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86 views

When can a metric space be embedded in the plane?

It's easy to check if a graph can be embedded in the plane: just check for forbidden minors. Is it also easy to check if a "distance function" can be embedded? Are there any necessary and sufficient ...
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1answer
99 views

Topology of a subset of continuous functions on the interval $[-1,1]$ on the metric space $(C[0,1],d_ { \infty})$

Problem statement Let $g \in C[-1,1]$. Consider the set $A=\{f \in C[-1,1] : f(x)\leq g(x), \space \forall x \in [-1,1]\}$. $a)$ Prove that on $(C[0,1],d_ {\infty})$, $A^ {\circ}=\{f \in C[-1,1], ...
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1answer
71 views

Prove this set is compact

Let $\{a_{n}\}_{n \in \mathbb{N}}$ be a sequence with the property that $\{a_{n}\}$ converges to $0$ when $n \rightarrow \infty$. Now let's consider this set: $$K=\{\{x_{n}\}_{n\in\mathbb{N}} \in ...
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1answer
58 views

operator from $\ell_1 $ to $\ell_1$ and density

I want to solve the following problem, but I am stuck... Let $a=(a_n)_n \in \ell_\infty $. Define the linear operator $ \displaystyle T : \ell_1 \to \ell_1 $ by $\displaystyle T(x) = a \cdot x , ...
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Continuous function from real line of the set of all $n \times n$ real matrix.

Let us take usual definitions of continuity on metric spaces and the usual distance metric on $M(n,\mathbb{R})$. I am looking for a continuous mapping $f : \mathbb{R} \rightarrow M(n, \mathbb{R})$, ...
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1answer
56 views

Prove statement about a sequence of homeomorphisms $f_n:\mathbb R \to \mathbb R$

The problem statement: Let $\{f_n\}_{n \in \mathbb N}$ be a sequence of homeomorphisms from $\mathbb R$ to $\mathbb R$ and let $F$ be a closed subset of $\mathbb R$ that doesn't contain any rational ...
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1answer
113 views

complete subset of a metric space

Let $f:X\to Y$ be a continuous map between metric spaces. Then $f(X)$ is a complete subset of $Y$ if A. the space $X$ is compact B. the space $Y$ is compact C. the space $X$ is complete D. the ...
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1answer
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Is a Banach space $X$ Lipschitz equivalent to the metric quotient $X/B$, where $B$ is the closed unit ball?

Recall that the metric quotient $X/B$ is defined as follows: first we consider the equivalence relation $\sim$ on $X$ that identifies all points of $B$, then we define on the set of all equivalence ...
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3answers
48 views

Prove that this series converges?

I have a Banach space $X$ and a linear operator $A \in L(X)$. $A$ is bounded such that $||A|| <1$. I then have to show that $$log(I-A)=\sum_{n \ge 1} \frac {A^n}n$$ converges. All I can come up ...
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1answer
97 views

Proving completeness and compactness of a sequence of metric spaces.

The problem statement Let $(X_n,d_n)_{n \in \mathbb N}$ be a sequence of metric spaces. Consider the product space $X=\prod_{n \in \mathbb N} X_n$ with the distance $d((x_n),(y_n))=\sum_{n \in ...
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1answer
87 views

Completeness of metric spaces

edit: Another question about formatting. My questions on math.stackexchange keep cutting off the last few lines of my post. How do I fix this? The remaining lines show up in my edit box but not in the ...
2
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1answer
48 views

Proving two statements about locally compact spaces

The problem statement: Let $(X,d)$ be a locally compact metric space (for every $x \in X$, there exists a compact neighbourhood of $x$) $a)$ Prove that if $K_1 \subset X$ is compact, then, there are ...
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1answer
53 views

Hypersphere isometry?

I will denote the $n-$sphere of radius $1$ centered at the origin as $\mathbb{S}^n$, so that $$ \mathbb{S}^n = \{ x \in \mathbb{R}^{n+1}\ : \ \|x\| = 1\}. $$ I am stuck on the following problem...I'm ...
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1answer
96 views

Prove that the space of divergent sequences in $(l_{\infty},d_{\infty})$ is open and dense. Is it separable?

The problem statements are: Consider the space $A=\{ \{a_n\}_{n \in \mathbb N} \in l_{\infty} : \{a_n\}_{n \in \mathbb N} \text { is not convergent }\}$ $a)$ Prove that $A$ is open and dense in ...
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1answer
47 views

Prove this sequence in $L(\ell_1)$ does not converge to zero?

I have a linear operator $$A: \ell_1 \rightarrow \ell_1$$ $$A_nx=(0,0,...,0,x_{n+1},x_{n+2},...)$$ with $n$ zeros. I am asked to show two things: that for any $x \in \ell_1$, $\lim_{n \rightarrow ...
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83 views

Proving a continuous function $f:K \cup A \to \mathbb R$ is uniformly continuous if $K$ is compact and $A$ is discrete.

Let $(X,d)$ be a metric space. Let $K \subset X$ compact and $A \subset X$: $\exists \delta>0$ such that $d(a,b)>\delta$ for all $a,b \in A$ with $a \neq b$. Consider in $K \cup A$ the induced ...
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Symmetrical endomorphisms and quadratic forms

(This last part of my linear algebra course is causing me quite a bit of headaches, so please be patient) Let $V$ be a vector space over the real field, and we'll indicate with $(\cdot,\cdot)$ its ...
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222 views

Topologically equivalent metrics but not strongly equivalent in$Lip_{M}(\mathbb{R})=\{f: [0,1] \rightarrow \mathbb{R} : |f(y)-f(x)|\leq M.|y-x| \}$

Let's consider this set $Lip_{M}(\mathbb{R})=\{f: [0,1] \rightarrow \mathbb{R} : |f(y)-f(x)|\leq M.|y-x| \}$ (i.e Lipschitz functions in $[0,1]$). How can I prove that $(Lip_{M}(\mathbb{R}), ...
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1answer
808 views

Is a set $U$ consisting of the single point $p$ open or closed?

I'm guessing here that $U$ would have to be closed, especially since for example the proof of the theorem that the union of two closed sets is closed is also valid if one of the sets is $U$. Still, ...
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1answer
46 views

Proving the set of “distance functions” on a compact set is a compact set itself

The problem statement. Let $(X,d)$ be a compact metric space and $C(X)=\{\phi: X \to \mathbb R : \phi \text{ is continuous}\}$. For each $x \in X$ we define the function $f_x: X \to \mathbb R$ ...
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1answer
60 views

Prove or disprove two statements about open functions on metric spaces

Let $f: (X,d) \to (Y,d')$ an open function (not necessarily continuous) between metric spaces. Decide whether the following statements are true or false: 1) If $A \subset X$ doesn't have isolated ...
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Uniform implies pointwise convergence

I had a question to show a sequence of functions $(x_n)$ in $C[0,1]$ (equipped with a metric $d$) does not contain a uniformly convergent subsequence. $$ x_n(t) = \ n(1-nt) \ \ \ \ \ \ \forall \ ...
4
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2answers
92 views

Show that A=$\{(x_1,…x_n) \in \Bbb R | -1\le x_1\le x_2\le …x_n\le 1\} \subset \Bbb R^n $ is closed.

The full question was: Show that A=$\{(x_1,...x_n) \in \Bbb R | -1\le x_1\le x_2\le ...x_n\le 1\} \subset \Bbb R^n $ is compact, but I was able to show correctly that it is bounded. However my ...