Proving that d is a metric on X
9355
If $d(x,y)$ is a metric, then $\frac{d(x,y)}{1 + d(x,y)}$ is also a metric
83159
Defining a metric space
104413
Is the exponential of $-d$ a positive definite kernel for a general metric space $(X, d)$?
95783
Fixed points of contractions in metric spaces
123039
If $0<\alpha < 1$ and $d(x,y)$ is metric, show that $((d(x,y))^{\alpha}$ is metric.
13848
Showing that if a subset of a complete metric space is closed, it is also complete
112329
Show that $d(z,a)=|\frac{z-a}{1-\overline{a}z}|$ is a metric on the unit disk
123753
In a metric $(X,d)$, prove that for each subset $A$, $x\in\bar{A}$ if and only if $d(x,A)=0.$
146967
Understanding equivalent metric spaces
119105
If $(X,d)$ is a complete metric space and $A$ is closed then show that for $x \in X$ there exists an element $a_0 \in A$ such that $d(x,A)=d(x,a_0)$
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
49374
Closure of a subset in a metric space
114061
Construct such $d$ that $(\mathbb{R} \setminus \mathbb{Z}, d)$ is complete metric space
108779
How to prove that this series is a metric: $d(x,y):=\sum_{i=0}^\infty \frac{|x_i -y_i|}{2^i (1+|x_i-y_i|)}$
117716
Condition on function $f:\mathbb{R}\rightarrow \mathbb{R}$ so that $(a,b)\mapsto | f(a) - f(b)|$ generates a metric on $\mathbb{R}$
140692
Existence of a complete metric on $(0,1)$
27517
Isometry in compact metric spaces