Simply connected space: Difference between revisions
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An arbitrary product of simply connected spaces is simply connected. This follows from the fact that the fundamental group of a product of path-connected spaces, is the product of their fundamental groups. | An arbitrary product of simply connected spaces is simply connected. This follows from the fact that the fundamental group of a product of path-connected spaces, is the product of their fundamental groups. | ||
{{retract- | {{retract-hereditary}} | ||
A retract of a simply connected space is simply connected. This follows from the fact that the fundamental group of a retract is a group-theoretic retract of the fundamental group of the whole space. | A retract of a simply connected space is simply connected. This follows from the fact that the fundamental group of a retract is a group-theoretic retract of the fundamental group of the whole space. | ||
Revision as of 21:15, 15 December 2007
This article defines a homotopy-invariant property of topological spaces, i.e. a property of homotopy classes of topological spaces
View other homotopy-invariant properties of topological spaces OR view all properties of topological spaces
This property of topological spaces is defined as the property of the following associated group: fundamental group having the following group property: trivial group
Definition
Symbol-free definition
A topological space is said to be simply connected if it satisfies the following equivalent conditions:
- It is path-connected, and any loop at any point is homotopic to the constant loop at that point
- It is path-connected, and its fundamental group is trivial
Definition with symbols
Fill this in later
Relation with other properties
Stronger properties
Weaker properties
- Semilocally simply connected space
- Simple space
- Space with Abelian fundamental group
- Space with perfect fundamental group
Metaproperties
Products
This property of topological spaces is closed under taking arbitrary products
View all properties of topological spaces closed under products
An arbitrary product of simply connected spaces is simply connected. This follows from the fact that the fundamental group of a product of path-connected spaces, is the product of their fundamental groups.
Retract-hereditariness
This property of topological spaces is hereditary on retracts, viz if a space has the property, so does any retract of it
View all retract-hereditary properties of topological spaces
A retract of a simply connected space is simply connected. This follows from the fact that the fundamental group of a retract is a group-theoretic retract of the fundamental group of the whole space.