Acyclicity is product-closed: Difference between revisions

Latest revision as of 02:19, 29 July 2011

This article gives the statement, and possibly proof, of a topological space property (i.e., acyclic space) satisfying a topological space metaproperty (i.e., product-closed property of topological spaces)
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Get more facts about acyclic space |Get facts that use property satisfaction of acyclic space | Get facts that use property satisfaction of acyclic space|Get more facts about product-closed property of topological spaces

Statement

For two spaces

Suppose $X_{1}$ and $X_{2}$ are topological spaces that are both acyclic spaces. Then, the product space $X_{1} \times X_{2}$ , endowed with the product topology, is also an Acyclic space (?).

For finitely many spaces

Suppose $X_{1}, X_{2}, \dots, X_{n}$ are topological spaces that are all acyclic spaces. Then, the product space $X_{1} \times X_{2} \times \dots \times X_{n}$ , endowed with the product topology, is also an acyclic space.

For an arbitrary number of spaces

Suppose $X_{i}, i \in I$ , are topological spaces that are all acyclic spaces. Then, the product space $\prod_{i \in I} X_{i}$ , endowed with the product topology, is also an acyclic space.

Related facts

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 ===For finitely many spaces===
-Suppose <math>X_1, X_2, \dots, X_n</math> are topological spaces that are all acyclic spaces. Then, the product space <math>X_1 \times X_2 \times \dots X_n</math>, endowed with the [[product topology]], is also an [[acyclic space]].
+Suppose <math>X_1, X_2, \dots, X_n</math> are topological spaces that are all acyclic spaces. Then, the product space <math>X_1 \times X_2 \times \dots \times X_n</math>, endowed with the [[product topology]], is also an [[acyclic space]].
 ===For an arbitrary number of spaces===