Algebraic Topology, Part 1: General Notions

Stuff

Some stuff.

Consider open sets \(U_1, U_2 \in X\). In general \(f(U_i)\) may not be open, however for an open map \(f\),

\[f(U_1 \cup U_2) = f(U_1) \cup f(U_2) \text{ is open.}\]

The Van Kampen theorem: If \(U_1\) and \(U_2\) are open sets, \(U_1 \cup U_2 = X\), and \(U_1 \cap U_2 = X\) is simply connected, then:

\[\pi_1(U_1, x_0) \ast \pi_1(U_2, x_0) \cong \pi_1(X, x_0).\]