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Tychonoff's theorem
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(Theorem)
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Let $(X_i)_{i\in I}$ be a family of nonempty topological spaces. The product space (see product topology) $$\prod_{i\in I}X_i$$ is compact if and only if each of the spaces $X_i$ is compact.
Not surprisingly, if $I$ is infinite, the proof requires the Axiom of Choice. Conversely, one can show that Tychonoff's theorem implies that any product of nonempty sets is nonempty, which is one form of the Axiom of Choice.
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"Tychonoff's theorem" is owned by matte. [ full author list (3) | owner history (2) ]
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See Also: compact
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Tichonov's theorem |
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Cross-references: implies, conversely, axiom of choice, proof, infinite, compact, product topology, product, topological spaces
There are 15 references to this entry.
This is version 8 of Tychonoff's theorem, born on 2002-01-01, modified 2003-04-11.
Object id is 1168, canonical name is TychonoffsTheorem.
Accessed 9043 times total.
Classification:
| AMS MSC: | 54D30 (General topology :: Fairly general properties :: Compactness) |
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Pending Errata and Addenda
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