Algebraic Topology · Problem 19

Cohomology — Allen Hatcher

Mathematics — Topology · direct-limits-countable-groups

Problem

Show that a direct limit of countable abelian groups over a countable indexing set is countable. Apply this to show that if

X

is an open set in

\mathbb{R}^n

then

H_i(X; \mathbb{Z})

is countable for all

i