Connected component

A connected component is a maximal connected subset of a topological space.

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.

An embedded copy of $S^{n-1}$ in ${\mathbb{R}}^n$ separates space into an inside and an outside. In particular, a connected closed surface embedded in ${\mathbb{R}}^3$ is two-sided and orientable.