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 spaces


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Definition

A topological space is said to be semilocally simply connected or semilocally 1-connected if it satisfies the following equivalent conditions:

  1. For any x \in X there exists an open subset U \ni x such that the homomorphism of fundamental groups induced by the inclusion of U in X is trivial (i.e., the image of the inclusion is the trivial subgroup), the inclusion being: \! \pi_1(U,x) \to \pi_1(X,x). Note that if U and/or X are not connected, we interpret the fundamental groups as referring to the fundamental groups of the path components of x in the respective subsets.
  2. For any x \in X and any open subset V of X containing x, there exists an open subset U \ni x such that U \subseteq V and the homomorphism of fundamental groups induced by the inclusion of U in X is trivial (i.e., the image of the inclusion is the trivial subgroup), the inclusion being: \! \pi_1(U,x) \to \pi_1(X,x). In other words, every loop about x contained in U, is nullhomotopic in X. Note that if U and/or X are not connected, we interpret the fundamental groups as referring to the fundamental groups of the path components of x in U and X 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 U is path-connected.

Relation with other properties

Stronger properties

References

Textbook references

  • Topology (2nd edition) by James R. MunkresMore info, Page 494 (formal definition)
  • An Introduction to Algebraic Topology (Graduate Texts in Mathematics) by Joseph J. RotmanMore info, Page 297 (formal definition): Introduced as semilocally 1-connected
  • Algebraic Topology by Allen HatcherFull text PDFMore info, Page 63 (formal definition)
  • Algebraic Topology by Edwin H. SpanierMore info, Page 78 (forma definition): Introduced as semilocally 1-connected