# 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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### I want to prove that $f$ is continuous if its graph is closed

This is an exercise from Rudin's 'Functional Analysis': Suppose that $X$ and $K$ are metric spaces, that $K$ is compact, and that the graph $f:X\to K$ is a closed subset of $X\times K$. Prove that ...
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### Open and connected set in metric space [duplicate]

In a normed space, we know that if a set is open and connected, it is path connected. Is it true for general metric space or general topological space?
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### distance between a real and R\Q

Please how to prove that $d(x, R\setminus Q)=0, d(x,Q)=0$ for all $x\in \mathbb{R}$ ? I know that $d(x,R\setminus Q)=\inf_{a\in R\setminus Q} d(x, a)$ but how to continue ? Can i say that As ...
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### What does it mean exactly by a metric “generates” a topology?

For example, the discrete metric $d(x,y)$ where $d(x,y) = 1$ if $x\neq y$, $d(x,y) = 0$ if $x = y$ "generates" the discete topology $\tau$ where $\tau = 2^X$ Can someone clarify exactly what is meant ...
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### On preimage of open sets of functions on real line having at most countably many discontinuity points

Let $f:\mathbb R \to \mathbb R$ be a function whose set of discontinuity points is at most countable ; is it true that for every open set $G \subseteq \mathbb R$ , there is an open set $U$ and a ...
Let $(X,d)$ be a compact metric space, and $f \colon (X,d)\to \mathbb{R}$ is a continuous function such that if $x,y \in X$ and $x \neq y$ than $f(x) \neq f(y)$. Let $t \colon X \times X \to ... 1answer 70 views ### what are the geodesics in the hyperbolic upper half plane? In the upper half-plane $$H = \{(x, y) \in \mathbb{R}^2 \mid y > 0\}$$ the distance between the two points (a,A) and (b,B) is set by the shortest curvature in metrix $$F(y) = \int_a^b ... 1answer 49 views ### Find a metric d on X such that (X, τ^{(d/X)}) is not connected X = (\{0\} \times [-1,1] \cup \{(x,\sin(π/x)) : x \in (0,1]\} \subset \mathbb{R}^2 Find a metric d on X such that (X, τ^{(d/X)}) is not connected. Note: τ^{(d/X)} denotes the metric ... 0answers 85 views ### Testing whether a particular set of measures borelianas is a set of Baire Let X compact metric space and F:X\times \mathbb{R}\rightarrow X flow continuous (F(x,t)=F_t(x)). If \delta>0 we define$$\Lambda(x,\delta)=\bigcup_{h\in ... 1answer 19 views ### Metric space with two similar points which are not in the same orbit. Is there an example of a metric space$X$with two points$p$and$q$so that for every$r>0$the ball with radius$r$and center$p$is isometric to the ball with radius$r$and center$q$and yet ... 2answers 51 views ### How can I prove that Let we have the following ultrametric space$(z,|.|_2)$such that if$x=r.2^n$then$|x|_2=2^{-n}$how can I prove that the topology produced by this metric isn't discrete topology ? 0answers 55 views ### If we think of infinity as a number, how does it affect the compactness/completeness of a metric space? I was recently reviewing some notes regarding compactness, in which the sequential definition is given i.e. "$A$is compact if any sequence in$A$has a subsequence which converges to a limit in$A$. ... 2answers 58 views ### A metric on the natural numbers Does there exist a complete metric on the set of natural numbers such that$\{n,\,n+1,\,n+2,\,\cdots\}$is a closed ball for each$n$? 0answers 52 views ### Mahalanobis distance Suppose there is a function$f$, for which we know the inequality $$f(r)\leq r$$ is true, where$r=||x-y||_2=\sqrt{(x-y)^T (x-y)}$is the Euclidean distance. If now we use the Mahalanobis distance ... 1answer 27 views ### Verification of proof of continuity between metric spaces and deduction from proof Let$M = [0,1]^{[0,1]}$and$d(f,g) = \sup{\{\lvert f(x) - g(x)\rvert \mid x \in [0,1]\}}$. For$a,b \in [0,1]$let$\phi_{a,b}(f) = f(b) - f(a)$($\phi$maps from$M \to \Bbb{R}$). Assume that ... 2answers 33 views ### compact metric spaces and infimum I am currently revising metric spaces and have come across a question which I am unable to answer and have no idea how to begin with. Let$(M,d)$be a compact metric space. Suppose$T \colon M \to ...
I've looked at a few questions online asking to determine the completeness of Metric Spaces. 2 such examples of metric spaces $(M,d)$: 1) $M = \{ (x,y) \in \mathbb{R}^2 \space : y>0$ or \$ x=0=y ...