Semilocally simply connected space: Difference between revisions

Revision as of 00:59, 28 January 2012

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:

For any $x \in X$ there exists an open subset $U ∋ x$ 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: $π_{1} (U, x) \to π_{1} (X, x)$ . Note that if $U < m a t h > a n d / o r < m a t h > X$ are not connected, we interpret the fundamental groups as referring to the fundamental groups of the path components of $x$ in the respective subsets.
For any $x \in X$ and any open subset $V$ of $X$ containing $x$ , there exists an open subset $U ∋ x$ 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: $π_{1} (U, x) \to π_{1} (X, x)$ . 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.

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 6: / Line 6: @@
 # For any <math>x \in X</math> there exists an open subset <math>U \ni x</math> such that the homomorphism of fundamental groups induced by the inclusion of <math>U</math> in <math>X</math> is trivial (i.e., the image of the inclusion is the trivial subgroup), the inclusion being: <math>\! \pi_1(U,x) \to \pi_1(X,x)</math>. Note that if <math>U<math> and/or <math>X</math> are not connected, we interpret the fundamental groups as referring to the fundamental groups of the [[path component]]s of <math>x</math> in the respective subsets.
-# For any <math>x \in X</math> and any open subset <math>V</math> of <math>X</math> containing <math>x</math>, there exists an open subset <math>U \ni x</math> such that <math>U \subseteq V</math> and the homomorphism of fundamental groups induced by the inclusion of <math>U</math> in <math>X</math> is trivial (i.e., the image of the inclusion is the trivial subgroup), 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>. Note that if <math>U<math> and/or <math>X</math> are not connected, we interpret the fundamental groups as referring to the fundamental groups of the [[path component]]s of <math>x</math> in <math>U</math> and <math>X</math> respectively.
+# For any <math>x \in X</math> and any open subset <math>V</math> of <math>X</math> containing <math>x</math>, there exists an open subset <math>U \ni x</math> such that <math>U \subseteq V</math> and the homomorphism of fundamental groups induced by the inclusion of <math>U</math> in <math>X</math> is trivial (i.e., the image of the inclusion is the trivial subgroup), 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>. Note that if <math>U</math> and/or <math>X</math> are not connected, we interpret the fundamental groups as referring to the fundamental groups of the [[path component]]s of <math>x</math> in <math>U</math> and <math>X</math> respectively.
 ==Relation with other properties==