Semilocally simply connected space

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

Definition

Symbol-free definition

A topological space is said to be semilocally simply connected or semilocally 1-connected if every point in the space has an open neighbourhood such that the inclusion map from that neighbourhood to the whoel space induces a trivial mapping at the level of fundamental groups.

Definition with symbols

A topological space $X$ is said to be semilocally simply connected if for any $x\in X$ there exists a neighbourhood $U$ of $x$ such that the homomorphism of fundamental groups induced by the inclusion of $U$ in $X$, is trivial, 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 $X$.

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