# Contractibility is product-closed

This article gives the statement, and possibly proof, of a topological space property (i.e., contractible space) satisfying a topological space metaproperty (i.e., product-closed property of topological spaces)

View all topological space metaproperty satisfactions | View all topological space metaproperty dissatisfactions

Get more facts about contractible space |Get facts that use property satisfaction of contractible space | Get facts that use property satisfaction of contractible space|Get more facts about product-closed property of topological spaces

## Contents

## Statement

### Property-theoretic statement

The property of topological spaces of being a contractible space, satisfies the metaproperty of topological spaces of being product-closed.

### Statement with symbols

Let , , be an indexed family of topological spaces. Then the product space, endowed with the product topology, is contractible.

## Proof

### Key idea (for two spaces)

Suppose and are contracting homotopies for and . Then the map defined as:

is a contracting homotopy for .

Thus is contractible.

### Generic proof (for an arbitrary family)

**Given**: An indexing set , a collection of contractible spaces. is the product of the s, endowed with the product topology

**To prove**: is a contractible space

**Proof**: Since each is contractible, we can choose, for each , a point , and a contracting homotopy , with the property that:

Now consider the point whose coordinate is for each . We denote:

to be a point whose coordinate is . Then, define a homotopy:

given by:

In other words, the homotopy acts as in each coordinate. We observe that:

- Since for each ,
- Since for each ,
- is a continuous map:
*Fill this in later*

Thus, is a contracting homotopy on , so is contractible.