Consider the metric space (Y,d), where d is the discrete metric. Find all connected subsets of (Y,d). Find all compact subsets of (Y,d).
As far as the definition of connectedness in metric spaces is concerned ie any connected metric space cannot be expressed as a union of disjoint non-empty open sets.I think that all single tons in the discrete metric space will form connected subsets. Am I correct?
However, I don't understand how to find compact subsets. Also, I think (X,d) will be complete.Am I correct?
discrete-space
Best Answer
Yes, you are correct with all those things.
Connected Components. Since with respect to the discrete every set is open and closed, in particular singletons are open and closed, so they form the connected components (because they themselves are connected).
Compact Subsets. The compact subsets are precisely the finite sets. To see this, suppose $K \subset X$ is compact. Since $X$ is discrete, the cover $\{\{x\} \mid x \in K\}$ is an open cover of $X$. Since $K$ is compact, it admits a finite subcover, so $K$ is a finite union of singletons. In other words, $K$ is finite.
Completeness. Completeness follows from the observation that a sequence is Cauchy in a discrete space if and only if it is eventually constant.