TITLE: Are perfect sets compact in R?
QUESTION [1 upvotes]: By the definitions of perfect and compact sets, are perfect sets, compact in $\mathbb{R}^n?$
What about in topological space?

REPLY [2 votes]: By definition, a set $A$ is perfect if $\text{der}(A)=A$ where $\text{der}$ denotes the set of accumulation points. $\mathbb{R}^n$ is perfect and non-compact. And clearly, if you take a non-compact topological space $X$ then it's perfect but non-compact.
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