Semilocally simply connected space

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

• Locally simply connected space
• Simply connected space

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 PDFMore info}, Page 63 (formal definition)
• Algebraic Topology by Edwin H. Spanier^{More info}, Page 78 (forma definition): Introduced as semilocally 1-connected