TITLE: Fundamental group of Alexandrov space.
QUESTION [5 upvotes]: Is it true that the fundamental group of a compact finite dimensional Alexandrov space with curvature bounded below is finitely generated?

REPLY [4 votes]: Under these hypotheses the systole of $X$ is clearly bounded from below, and the usual comparison arguments would give an upper bound on the number of points in an $\epsilon$-net in $X$.  Then every loop can be discretized and the finite-generation follows.