simply connected

A topological space is said to be simply connected if it is path connected and the fundamental group of the space is trivial (i.e. the one element group). What this means, basically, is that every path on the space can be shrunk to a point. This is equivalent to saying that every path is contractible. A simply connected space can be visualized as a space with no “holes”.

Some basic examples of a simply connected space are the unit disc in $\\mathbb{R}^{2}$ , $S^{2}$ or the Riemann sphere. Non-examples of a simply connected space are the circle, the annulus, and a punctured plane (a plane with a point removed). In each of the non-examples, any closed curve around the “hole” is a path that can not be shrunk to a point.

单词	SimplyConnected
释义	simply connected A topological space is said to be simply connected if it is path connected and the fundamental group of the space is trivial (i.e. the one element group). What this means, basically, is that every path on the space can be shrunk to a point. This is equivalent to saying that every path is contractible. A simply connected space can be visualized as a space with no “holes”. Some basic examples of a simply connected space are the unit disc in $\\mathbb{R}^{2}$ , $S^{2}$ or the Riemann sphere. Non-examples of a simply connected space are the circle, the annulus, and a punctured plane (a plane with a point removed). In each of the non-examples, any closed curve around the “hole” is a path that can not be shrunk to a point.
随便看	CountablyCompact countably compact CountablyInfinite countably infinite Counterexample counterexample CounterexampleOfFubinisTheoremForTheLebesgueIntegral counter-example of Fubini’s theorem for the Lebesgue integral CounterexamplesForProductsAndCoproduct counterexamples for products and coproduct CounterExampleToNakayamasLemmaForNonfinitelyGeneratedModules counter example to Nakayama’s lemma for non-finitely generated modules CounterexampleToTonellisTheorem counter-example to Tonelli’s theorem CountingComplexityClass counting complexity class CountingCompositionsOfAnInteger counting compositions of an integer CountingMeasure counting measure CountingProblem counting problem CountingProcess counting process CountingTheorem