Find an example of a complete metric space $X$ in which all sufficiently small closed balls are compact but large ones are not.
First of all, what is ''a small ball'' and ''a large ball'' in a metric space? Does it have to do with the radius of the ball?
Theorem: Every closed ball of $X$ is compact $\Leftrightarrow$ $X$ has the nearest-point property.
Since $X$ has not the nearest-point property, it is not compact. But $X$ is complete so the space is not total bounded.
I think this is the fact that we want to take advantage of, to prove that large balls are not compact.
Not sure how to proceed or if this thinking is correct.
The nearest-point property is not in many topology books, so i give the definition:
$(X,d)$ is a metric space. The following are equivalent:
i) Every infinite bounded subset of $X$ has an accumulation point in X(BW criterion)
ii) Every bounded sequence of $X$ has a convergence subsequence in $X$.
iii) X is complete and every bounded subset of $X$ is totally bounded.
Best Answer
Let $X$ be an infinite discrete space with the usual discrete metric, $d(x,y)=1-\delta_{xy}$. Since $X$ is infinite and discrete $\{x\}_{x\in X}$ is an open cover with no finite subcover. Cauchy sequences in $X$ are eventually constant, and thus converge, and any sufficiently small closed ball contains a single point and is thus compact. Note that $X$ is the closed ball of radius 1 around any point. Hence $X$ satisfies your conditions.
Edit: Some comments on my thought process, in response to the question in the comments.
My first thought was that we know that a metric space is compact if and only if it is complete and totally bounded. Since our whole metric space is complete here, we need to find a way to make small closed balls totally bounded and large balls not totally bounded. The way I think about total boundedness, this is similar to saying we want small neighborhoods of a point to be "finite dimensional" and large neighborhoods to be "infinite dimensional." In a very rough intuitive way.
From there, the discrete topology is fairly natural, since small balls are literally finite and large balls contain the whole space.
As for my thoughts on your thoughts, well, they're correct, but I'm not sure where you would go from where you got to.