Let be $\rho (x,y) $ a metric on X. Prove that $\rho _1 (x,y)=\rho (x,y)/(1+\rho (x,y) ) $ is a metric on X.
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?
74607
Variation of the metric with respect to the metric
119247
Let $(M,d)$ be a compact metric space and $f:M \to M$ such that $d(f(x),f(y)) \ge d(x,y) , \forall x,y \in M$ , then $f$ is isometry?
27517
Isometry in compact metric spaces
39763
Proving left-invariance (and proof-verification for right-invariance) for metric constructed from left-invariant Haar measure
90628
The metric spaces are Tychonoff spaces
15365
Proving that the triangle inequality holds for a metric on $\mathbb{C}$
133504
Suppose $X$ is a metric space and $S \subseteq X.$ Then, $S^o=\{x \in X~|~dist(x,S^c)>0\}$.
5042
Order of a product of subgroups. Prove that $o(HK) = \frac{o(H)o(K)}{o(H \cap K)}$.
66531
$\|h(x)-h(y)\|\geq 3\|x-y\|$ prove that image of open is open
105437
How can I show these two metrics give the same topology?
23224
$\mbox{rank}(A)=1$ implies $\det(A+I)=\mbox{trace}(A)+1$
85242
Deciding whether two metrics are topologically equivalent in the space $C^1([0,1])$
128598
How to prove this is a metric?
94571
How can we prove that $N_G(H)=H$ implies $N_{f(G)}(f(H))=f(H)$
83273
Show that $o(gHg^{-1})=o(H)$