Which complete metric space has an infinite compact subset?

Question

In any complete metric space $X$ (infinite), a finite set is compact. Hence we go on to study the existence of infinite compact subsets.

Work:

1. Obvious examples of $X$ are usually a path-connected subset of a vector space,

e.g. scalar multiple of a vector ($c*g$, Which $g$ is a function in $C[0,1]$), surfaces.

Their infinite compact subsets exist by considering continuous functions $f$ to the image set, hence the image must be compact.

2. By considering infinite subsets of $X$,

if $X$ has an infinite compact subsets, $X$ then has an infinite subset that has a limit point.

Now if all infinite subsets of $X$ have no limit points,

consider $X$ as subset of $X$, since $X$ also doesn't have any limit point, hence for every $x \in X$, there is a $\epsilon$-ball ($\epsilon$ may not be uniform) of $x$, whose intersection with $X$ is $\{x\}$ only, hence is isolated.

Hence $X$ is a discrete set, my question is to find interesting sets that have no infinite compact subset.

Obvious example:

1. $\mathbb{N}$ under $d(x,y)=|x-y|$

2. discrete subsets of $\mathbb{R}^n$

Added:

1. From both solution, it is known that completeness is not required.

2. It is known that interesting discrete subsets are generally not interesting.

Answer 1

Suppose that $X$ is a non-discrete metric space. Then there is a subset $A$ of $X$ which is not closed. Let $x\in\overline A\setminus A$. Then $x=\lim_{n\to\infty}a_n$, for some sequence $(a_n)_{n\in\mathbb N}$ of elements of $A$. Clearly, the set $\{a_n\mid n\in\mathbb N\}$ cannot be finite; otherwise, $x\in\{a_n\mid n\in\mathbb N\}\subset A$. So,$$\{x\}\cup\{a_n\mid n\in\mathbb N\}$$is an infinite compact subset of $X$.

Answer 2

If $X$ has any infinite set $T$ with a limit point $t$, then an injective sequence in $T$ converging to $t$ is a homeomorphism from $\{1/n:n\in\mathbb N\}\cup \{0\}$ to a subspace of $X$. Thus a metric space admits an infinite compact subset if and only if it admits a(n infinite) subset with a limit point, if and only if it is not discrete. Note that completeness is not relevant here.