Fundamental group of product is product of fundamental groups

Statement

For two based topological spaces

Suppose $(X,x_{0})$ and $(Y,y_{0})$ are based topological spaces. Then, the following is true for the fundamental groups of the topological spaces $(X,x_{0})$ , $(Y,y_{0})$ and the product space $(X\times Y,(x_{0},y_{0}))$ :

$\!\pi _{1}(X\times Y,(x_{0},y_{0}))\cong \pi _{1}(X,x_{0})\times \pi _{1}(Y,y_{0})$

More explicitly, if $p_{X}$ and $p_{Y}$ denote the projections from $(X\times Y,(x_{0},y_{0}))$ to $(X,x_{0})$ and $(Y,y_{0})$ respectively, then the maps:

$\!\pi _{1}(p_{X}):\pi _{1}(X\times Y,(x_{0},y_{0}))\to \pi _{1}(X,x_{0})$

and:

$\!\pi _{1}(p_{Y}):\pi _{1}(X\times Y,(x_{0},y_{0}))\to \pi _{1}(Y,y_{0})$

then under the isomorphism $\pi _{1}(X\times Y,(x_{0},y_{0}))\cong \pi _{1}(X,x_{0})\times \pi _{1}(Y,y_{0})$ we get the direct factor projections for the group product.

For two topological spaces without basepoint specification

Suppose $X$ and $Y$ are both path-connected spaces, or more generally, each of them is a space such that all the path components of the space are homeomorphic to each other. Then, the fundamental groups $\pi _{1}(X)$ , $\pi _{1}(Y)$ , and $\pi _{1}(X\times Y)$ are all well-defined without specification of basepoint (See fundamental group#Omission of basepoint). We then have: