Semilocally simply connected space: Difference between revisions

Revision as of 21:54, 21 April 2008

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

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

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

@@ Line 5: / Line 5: @@
 ===Symbol-free definition===
-A [[topological space]] is said to be '''semilocally simply 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.
+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]] <math>X</math> is said to be '''semilocally simply connected''' if for any <math>x \in X</math> there exists a neighbourhood <math>U</math> of <math>x</math> such that the homomorphism of fundamental groups induced by the inclusion of <math>U</math> in <math>X</math>, is trivial. In other words, every loop about <math>x</math> contained in <math>U</math>, is nullhomotopic in <math>X</math>.
+A [[topological space]] <math>X</math> is said to be '''semilocally simply connected''' if for any <math>x \in X</math> there exists a neighbourhood <math>U</math> of <math>x</math> such that the homomorphism of fundamental groups induced by the inclusion of <math>U</math> in <math>X</math>, is trivial, the inclusion being:
+<math>\pi_1(U,x) \to \pi_1(X,x)</math>
+In other words, every loop about <math>x</math> contained in <math>U</math>, is nullhomotopic in <math>X</math>.
 ==Relation with other properties==
@@ Line 17: / Line 21: @@
 * [[Locally simply connected space]]
 * [[Simply connected space]]
+==References==
+===Textbook references===
+* {{booklink|Munkres}}, Page 494 (formal definition)
+* {{booklink|Rotman}}, Page 297 (formal definition): Introduced as '''semilocally 1-connected'''
+* {{booklink|Hatcher}}, Page 63 (formal definition)
+* {{booklink|Spanier}}, Page 78 (forma definition): Introduced as '''semilocally 1-connected'''