# products of connected spaces are connected

###### Theorem 1

[1, 2] Let $(X_{i})_{i\in I}$ be a family of topological spaces. Then the product space

 $\prod_{i\in I}X_{i}$

with the product topology is connected if and only if each space $X_{i}$ is connected.

As is true of most results in topology involving products, the forward implication requires the axiom of choice.

## References

• 1 S. Lang, Analysis II, Addison-Wesley Publishing Company Inc., 1969.
• 2 A. Mukherjea, K. Pothoven, Real and Functional Analysis, Plenum Press, 1978.
Title products of connected spaces are connected ProductsOfConnectedSpacesAreConnected 2013-03-22 13:56:13 2013-03-22 13:56:13 mps (409) mps (409) 6 mps (409) Theorem msc 54D05