# Tagged Questions

Metric spaces are sets on which a metric is defined. A metric is a generalisation of the concept of "distance". Metric spaces should not be confused with topological spaces.

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### $A,B$ be countable dense subsets of $\mathbb R$ , let $A,B$ be given usual subspace topologies , then there exists a homeomorphism $f:A \to B$?

Let $A,B$ be countable dense subsets of $\mathbb R$ (with usual euclidean topology ) let $A,B$ be given usual subspace topologies , then is it true that there exists a homeomorphism $f:A \to B$ ?
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### Show that if $(X, \mathcal{T}), (Y, \mathcal{J})$ are both metrizable, then $X \times Y$ with product topology is metrizable

Let $(X, \mathcal{T}), (Y, \mathcal{J})$ be both metrizable, then Claim: $X \times Y$ with product topology is metrizable I have made an attempt at this question but the notation is ...
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### Pseudometric without triangle inequality

I'm working on an optimization problem where I aim to minimize the total euclidean distance of the edges of a graph drawn on a fixed-size grid. For convenience, I actually use the maximum of $0$ and ...
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### Weak measurability of a set-valued map

Suppose that $A$ and $B$ are compact metric spaces. Let $f:A\times B\to B$ be a Borel measurable map (in the sense that for every Borel set $S\subseteq B$, $f^{-1}(S)$ belongs to the $\sigma$-algebra ...
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### The compactness of metric space $\mathcal{G}_n$

Here, metric space $\mathcal{G}_n$ is described below: It is said that it is well-known that $\mathcal{G}_n$ is compact for every $n$. But I can't find a proof, can you give me a proof or an ...
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### Function from space of continuous functions to reals is continuous (Proof Verification)

Question: $C$ is the space of continuous functions from $[0,1]$ to $\mathbb{R}$ under the sup metric. Prove the function $$f:C\to\mathbb{R}\quad f\to \int_0^1 f(t)^2 dt$$ is continuous. My answer: ...
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### A set is compact iff every collection… Proof check

I asked this question (A set is compact iff all closed collections of subsets with the f.i.p. have nonempty intersection) a few days ago and was lucky enough to get an answer, but I'm afraid that the ...
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### Help understanding a proof that the metric space of bounded functions is complete.

Theorem Let $M$ and $N$ be metric spaces, then the metric space $F_{b}(M,N)$ $= \{f:M \to N : \text{f bounded} \}$ is complete if $N$ is complete Proof Let $\{f_k\}_{k=1}^{\infty}$ be a Cauchy ...
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### Proof of the Beltrami theorem

I'm studying from a textbook and came across a theorem as the following, which it calls the Beltrami Theorem: Beltrami Theorem: A Riemannian metric $g$ is projectively flat if and only if it is ...
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### Does a mapping from one metric space to another metric space preserve star-likeness of regions?

Let $X$ be a vector space, let $M_1 = \left(X, d_1\right)$ be a metric space and let $M_2 = \left(X, d_2\right)$ be another one. $f : M_1 \to M_2$ is continuous and the origin is a fixed point. $f$ ...
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### Definition of locally connected metric space

I have this definition of locally connected metric space: "A metric space $(X,d)$ is called locally connected if for all $x\in X$ and for all $U\subset X$, $U$ neighbourhood of $x$, exists a connected ...
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### Left invariant metrics on a Lie group coming from Lie algebras

Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra. If we have an inner product on $\mathfrak{g}$ then we can create a left invariant metric $d_G$ on $G$ by translations. On the other hand ...
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### Compact convergence of inverse functions

Consider two metric spaces $X$ and $Y$ and a sequence of functions $f_n\colon X\to Y$ together with a function $f\colon X\to Y$. Assume, all $f_n$ and $f$ have inverse functions $g_n$ and $g$, say. It ...
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### Find a metric on the simplex so that every transposed positive stochastic matrices becomes a contraction.

A stochastic matrix $P$ is a $n \times n-$matrix with entries $p_{ij} \in [0,1]$ so that $\sum_{k=1}^n p_{ik} = 1$ for every $i \in \{1,...,n\}$. The matrix $P$ is called positive, if no entry $p_{ij}$...
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### Understanding a theorem of Dynamical systems, mostly definitions

I am reading, the following paper for my research: http://www.sciencedirect.com/science/article/pii/S0022247X00973438 I need to know what are the precise definitions of the following terms, since I ...
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### completion of a complete metric space is itself

Suppose $(X,d)$ is a complete metric space and $(Y,d)$ is its completion . What can be said from this $?$. I think $Y=X$ is the answer . Am I correct $?$
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### Proof that the function on $C[0,b]$ is a contraction

Question : Let $a$,$b$ be real numbers with $0\lt b\lt 1$ . Consider the subset $X\subset C[0,b]$ consisting of the functions $f$ s.t $f(0)=a$ .Then $X$ is closed in $C[0,b]$. ...
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### Definition of a length metric.

Let $(X, d)$ be a metric space. Define the induced intrinsic metric $\widehat{d}$ as follows \widehat{d}(x,y) = \inf_{\gamma} \sup_{t_0, \ldots, t_N} \sum_{i=1}^{N}d(\gamma(t_{i-1}, \...
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### Proof that the closed interval in $\mathbb{R}$ is connected

Let $C$ be an open and closed subset of $[a,b]$. WLOG, assume $a \in C$. Set $A = \{x \in [a,b]: [a,x] \subseteq C\}$. Since $a \in A$, sup$A$ exists. Let $\epsilon > 0$. Then, (from real analysis)...
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### What does it mean when two sets are “adjoined” in a metric space?

I encountered the word "adjoined" in Baby Rudin, Chapter 2 concerning basic topology on Euclidean space. It appeared in the proof to Theorem 2.35 Theorem$\quad$ Closed subsets of compact sets are ...
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### Contraction principle for Fredholm integral equation of the first kind

A Fredholm integral equation of the first kind has the following shape: $$\int a(x,y)f(y)\mathrm dy = b(x)\tag{1}$$ where $f$ is an unknown function. I wonder whether contraction principle can be ...
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### Ball with euclidean metric - mistake in book?

In $\mathbb{R}^2$, the ball with euclidean metric $d_{l_2}$ is defined, in terence tao's analysis vol II, as: $$B_{(\mathbb{R}^2,d_{l_2})}((0,0),1) = \{(x,y) \in \mathbb{R}^2 : x^2 + y^2 <1\}$$ ...
Let $\mathcal{X}\equiv\Bbb{R}^n\times\Bbb{S}_{++}^n$, where $\Bbb{S}_{++}^n$ is the space of all symmetric positive-definite $n\times n$ real matrices. Let $x,y\in\mathcal{X}$, where $$x=(\mathbf{x},... 0answers 63 views ### Hölder's inequality/Cauchy-Schwarz for Bregman Divergence? Consider the Bregman divergence.$$ D_F(p, q) = F(p)-F(q)-\langle \nabla F(q), p-q\rangle.  And its dual norm: $D_{F*}(p, q)$ where $F^*(y) = \arg\min_x \left\{ \langle x, y\rangle - F(x) \... 0answers 32 views ### Any compact metric space is Borel equivalent to some subset of$[0, 1]$In Petersen's Riemannian Geometry book I encountered the following statement : Any compact metric space$X$is Borel equivalent to some$S \subset [0, 1]$i.e. there is a bijection$f : X \rightarrow ...
Is it true that every subset $Y$ of a metric space $X$ is a normal topological space? I think the answer is yes, because $Y$ is a metric subspace of $X$ equipped with the induced metric by the one of ...