How to check the following equivalent description of the compactness of metric space?
19111
If every real-valued continuous function is bounded on $X$ (metric space), then $X$ is compact.
85242
Deciding whether two metrics are topologically equivalent in the space $C^1([0,1])$
52014
Showing $(C[0,1], d_1)$ is not a complete metric space
45581
Is the distance function in a metric space (uniformly) continuous?
121218
Is this function a metric?
11114
Is a closed subset of a compact set (which is a subset of a metric space $M$) compact?
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 $\epsilon>0$
20164
Is there always an equivalent metric which is not complete?
3537
Show the Euclidean metric and maximum metric are strongly equivalent.
39076
Contraction of compact sets
50677
Is this metric space incomplete?
49374
Closure of a subset in a metric space
49563
Cauchy nets in a metric space
150987
Showing that the product and metric topology on $\mathbb{R}^n$ are equivalent
26794
Metrizability of Hausdorff continuous image of compact metric space
35652
Every compact metric space is complete