Semilocally simply connected space
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This article defines a homotopy-invariant property of topological spaces, i.e. a property of homotopy classes of topological spacesView other homotopy-invariant properties of topological spaces OR view all properties of topological spaces
Contents
Definition
A topological space is said to be semilocally simply connected or semilocally 1-connected if it satisfies the following equivalent conditions:
- For any there exists an open subset such that the homomorphism of fundamental groups induced by the inclusion of in is trivial (i.e., the image of the inclusion is the trivial subgroup), the inclusion being: . Note that if and/or are not connected, we interpret the fundamental groups as referring to the fundamental groups of the path components of in the respective subsets.
- For any and any open subset of containing , there exists an open subset such that and the homomorphism of fundamental groups induced by the inclusion of in is trivial (i.e., the image of the inclusion is the trivial subgroup), the inclusion being: . In other words, every loop about contained in , is nullhomotopic in . Note that if and/or are not connected, we interpret the fundamental groups as referring to the fundamental groups of the path components of in and respectively.
Note that the term is typically used for spaces that are locally path-connected spaces. In this case, we can assume that the open subset is path-connected.
Relation with other properties
Stronger properties
References
Textbook references
- Topology (2nd edition) by James R. Munkres^{More info}, Page 494 (formal definition)
- An Introduction to Algebraic Topology (Graduate Texts in Mathematics) by Joseph J. Rotman^{More info}, Page 297 (formal definition): Introduced as semilocally 1-connected
- Algebraic Topology by Allen Hatcher^{Full text PDF}^{More info}, Page 63 (formal definition)
- Algebraic Topology by Edwin H. Spanier^{More info}, Page 78 (forma definition): Introduced as semilocally 1-connected