PlanetMath: products of connected spaces are connected

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products of connected spaces are connected

(Theorem)

Theorem 1 [1,2]

Let $(X_i)_{i\in I}$ be a family of topological spaces. Then the product space$$\prod_{i\in I}X_$$ 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.

Bibliography

1: S. Lang, Analysis II, Addison-Wesley Publishing Company Inc., 1969.
2: A. Mukherjea, K. Pothoven, Real and Functional Analysis, Plenum Press, 1978.

"products of connected spaces are connected" is owned by mps. [ full author list (3) | owner history (2) ]

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Attachments:: proof that products of connected spaces are connected (Proof) by yark

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Cross-references: axiom of choice, connected, product topology, product, topological spaces

This is version 3 of products of connected spaces are connected, born on 2003-09-05, modified 2004-02-18.
Object id is 4697, canonical name is ProductsOfConnectedSpaces.
Accessed 2053 times total.

Classification:

54D05 (General topology :: Fairly general properties :: Connected and locally connected spaces )

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