products of connected spaces are connected
Theorem 1
[1, 2] Let ${\mathrm{(}{X}_{i}\mathrm{)}}_{i\mathrm{\in}I}$ 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 |
---|---|
Canonical name | ProductsOfConnectedSpacesAreConnected |
Date of creation | 2013-03-22 13:56:13 |
Last modified on | 2013-03-22 13:56:13 |
Owner | mps (409) |
Last modified by | mps (409) |
Numerical id | 6 |
Author | mps (409) |
Entry type | Theorem |
Classification | msc 54D05 |