Prove complete metric space for $I=]0,\infty[$ with $d(x,y)=\lvert\ln(x)-\ln(y)\rvert$
51227
Proving that given metric space is complete: $X := (0,\infty)$ and $d:=|\ln(x)-\ln(y)|$
133504
Suppose $X$ is a metric space and $S \subseteq X.$ Then, $S^o=\{x \in X~|~dist(x,S^c)>0\}$.
107701
Show that a connected metric space is $\epsilon$-chainable for $\epsilon>0$
82333
let $(X,d)$ be a metric space. How I can show that any finite subset of $X$ is closed.
118328
Metric space is totally bounded iff every sequence has Cauchy subsequence
146967
Understanding equivalent metric spaces
96074
A net $\varphi : [0, \omega_1) \to M$ on a metric space $M$ converges $\iff \varphi$ is eventually constant
3751
Proving a distance between molecules defines a metric space.
26547
Real numbers equipped with the metric $ d (x,y) = | \arctan(x) - \arctan(y)| $ is an incomplete metric space
41298
Understanding proof of completeness of $L^{\infty}$
133152
(X,d) be a compact metric space. For every open cover, show there exists ε > 0 such that ∀𝑥 ∈ X, B(x,ε) is contained in some member of the cover.
7402
Showing a compact metric space has a countable dense subset
55016
Prove that if ($d(x,y)=0$ iff $x=y$) and if $d(x,z)\leq d(x,y)+d(z,y)$ then $d$ is a metric
121359
Show that $d_b(x,y)=\frac{d(x,y)}{1+d(x,y)}$ is a metric.
35291
How to show $d(x,y)= \sqrt{d_1(x_1,y_1)^2+d_2(x_2,y_2)^2}$ is a metric?
140692
Existence of a complete metric on $(0,1)$
124179
Proof that every metric space is normal