Connected components of a product space
= Connected components of a product space
{title2=$C_{(x,y)}=C_x\times C_y$}
The <product of connected spaces> is connected, so the product of the two <connected components> lies in the component of $(x,y)$. Conversely, continuous projections carry that component to connected sets in the factors, which must lie in $C_x$ and $C_y$. These two inclusions prove equality in arbitrary <topological spaces>.