# Questions tagged [metric-spaces]

Metric spaces are sets on which a metric is defined. A metric is a generalization of the concept of "distance" in the Euclidean sense. Metric spaces arise as a special case of the more general notion of a topological space.

10,260 questions
36 views

### Metric Space defined by an Infinite Sequence of Metric Spaces in this case not a Metric Space

I just started my first book on Topology, Introduction to Topology by Bert Mendelson (Second Edition), and the second chapter is about Metric Spaces. The last problem of Section 2, Problem 2.9, is as ...
24 views

### Connected metric space of first category

Question: Does there exist a connected metric space of first category (i.e. it can be written as a countable union of nowhere dense subsets)? No such example is given in the book Counterexamples ...
38 views

29 views

### Example of a space that is separable but not complete

I know that in general metric space $X$ can be separable without being complete. What's a good example?
9 views

### Prove or disprove $bd(E)$ is nowhere dense for $E \subseteq X$ complete metric space

I know this is not true but need to find an example of complete metric space $X$ with subset $E$ such that $\overline{bd(E)}$ has non-empty interior.
Let $f,g : (X, \rho) \longrightarrow (Y,d)$ be continuous functions and $D$ a dense subset of $X$. Show that, if $f|_D = g|_D$ then $f=g$, where $\cdot |_D$ is the restriction to $D$. Let $w = f -... 2answers 53 views ### Can such$f$be continuous at the boundary? The given problem says , Let$f$be a complex holomorphic on the open unit disk$D$such that$|f(z)|\longrightarrow1$as$|z| \longrightarrow 1$, and$f$is nonzero inside the open unit disk.Can ... 0answers 34 views ### Which$[0,\infty]$-categories are compact and connected metric spaces? The title says it all. Which$[0,\infty]$-categories, regarded as generalized metric spaces, correspond to connected metric space? Which categories correspond to compact spaces? (I have a guess for ... 0answers 29 views ### The study of Hausdorff distance as a pseudometric. In hyperspace theory, if$(X,d)$is a metric space, then the Hausdorff distance between nonempty subsets$A$and$B$of$X\$ can be defined as $$H(A,B)=\text{max}\{\text{sup}\{d(a,B): a\in A\},\text{... 1answer 28 views ### A continuity problem \textbf{Problem:} Let f:(X,d_1)\rightarrow (Y,d_2) a continuous function and B \subseteq Y. Consider the set :$$ A =\{ x \in X \vert d_2(f(x),Y-B)>0 \} $$Prove that : \forall x \in A :... 0answers 52 views ### Compact + chain-connected metric space => connected How to show that if M (a metric space) is compact and chain-connected then it is connected ? Definition of \varepsilon-chainable : (X,d) is \varepsilon-chainable if given any two points a,b\... 0answers 23 views ### Heuristic argument for pointwise convergence imply uniform convergence I didn't want put the statement precisely in the title of the topic to not have a long title, but the items below describe precisely when pointwise convergence imply uniform convergence: Every ... 2answers 140 views ### Find bijection \ell^{\infty} \to [0,1] In a comment on the first answer to this question, @Nate Eldredge stated that "For instance, there is a metric on \ell^{\infty} that makes it isometric to [0,1]. How would that look like? ... 1answer 29 views ### Example on proving continuity in metric spaces Prove that f(x)=x+y^2+xy is continuous as a function \mathbb{R}^2\rightarrow \mathbb{R} in the metric d_1(x,y)=|x_1-y_1|+|x_2-y_2|. (assume that on \mathbb{R} we consider the usual absolute ... 1answer 18 views ### Question on completing proof separability in metric space implies countability [duplicate] Let X be a metric space. I want to show that: X separable \Rightarrow X second countable. What I have been able to show thus far: Let Y be the countable dense subset in X. I have shown ... 2answers 51 views ### Two analysis problems I am doing these two problems to review my Analysis classes. (a) Show that for each n \geq 1, there exists an x > 0 such that \frac{1}{\sqrt{nx+1}} + \frac{1}{\sqrt{nx+2}} + \... 0answers 24 views ### Intuition on an isometry between the taxicab and max metric in \mathbb{R}^2. I was curious when someone suggested a rotation of \pi/4 and a rescaling by \sqrt{2} gives a nice isometry between the taxicab and max metric on \mathbb{R}^2. Thus, let$$f: (x,y) \mapsto (\frac{...
Let $$f:X\rightarrow Y,$$ where X and Y are metric spaces and Y is compact. If f has a closed graph $$G_f=\{(x,f(x)):x\in X\},$$ then f is continuous. My first instinct is to use the fact that if ...