### What does **classifying space** mean?

# Definitions for classifying space

clas·si·fy·ing space

#### This dictionary definitions page includes all the possible meanings, example usage and translations of the word **classifying space**.

### Wiktionary

classifying spacenoun

A topological space that is the quotient of a free action (of the specified group) on a weakly contractible space.

### Wikipedia

Classifying space

In mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG (i.e. a topological space all of whose homotopy groups are trivial) by a proper free action of G. It has the property that any G principal bundle over a paracompact manifold is isomorphic to a pullback of the principal bundle EG → BG. As explained later, this means that classifying spaces represent a set-valued functor on the homotopy category of topological spaces. The term classifying space can also be used for spaces that represent a set-valued functor on the category of topological spaces, such as Sierpiński space. This notion is generalized by the notion of classifying topos. However, the rest of this article discusses the more commonly used notion of classifying space up to homotopy. For a discrete group G, BG is, roughly speaking, a path-connected topological space X such that the fundamental group of X is isomorphic to G and the higher homotopy groups of X are trivial, that is, BG is an Eilenberg–MacLane space, or a K(G,1).

### Wikidata

Classifying space

In mathematics, specifically in homotopy theory, a classifying space BG of a topological group G is the quotient of a weakly contractible space EG by a free action of G. It has the property that any G principal bundle over a paracompact manifold is isomorphic to a pullback of the principal bundle EG → BG. For a discrete group G, BG is, roughly speaking, a path-connected topological space X such that the fundamental group of X is isomorphic to G and the higher homotopy groups of X are trivial, that is, BG is an Eilenberg-Maclane space, or a K.

### Numerology

Chaldean Numerology

The numerical value of classifying space in Chaldean Numerology is:

**7**Pythagorean Numerology

The numerical value of classifying space in Pythagorean Numerology is:

**6**

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"classifying space." *Definitions.net.* STANDS4 LLC, 2024. Web. 15 Aug. 2024. <https://www.definitions.net/definition/classifying+space>.

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