Wedge sum

Remarks

For two spaces one writes $X \vee Y$. Each $X_i$ embeds in the wedge sum, and a pointed map out of $\bigvee_i X_i$ is the same as a family of pointed maps out of the $X_i$. Example: $S^1 \vee S^1$ is homeomorphic to the figure eight $\set{z \in \C : \abs{z - 1} = 1 \text{ or } \abs{z + 1} = 1}$, where the two circles meet only at $0$. Non-example: the disjoint union $S^1 \sqcup S^1$ without collapsing the base points is not a wedge sum; it is disconnected, while the wedge of connected spaces is connected. The wedge sum depends on the choice of base points in general, but not up to homotopy for well-behaved spaces such as connected CW complexes.