Alphabeta MathHow statement and proof provenance work
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Simply connected topological spaces
Definition
A topological space is simply connected when it is nonempty and path-connected (Paths, path-connected spaces and path components) and, for every , the group has exactly one element.
Requiring every basepoint avoids presuming a change-of-basepoint theorem. For a path-connected space that later theorem shows that checking one basepoint is equivalent, but no such result is needed for this definition. The empty space is path-connected under the published convention, but it is not simply connected here because nonemptiness is explicit.
Depends on
Used by
- A connected covering of a locally path-connected simply connected space is one-sheeted and trivial Corollary
- A simply connected overlap turns the van Kampen pushout into a free product Corollary
- If one set in a van Kampen cover is simply connected, the other fundamental group surjects with overlap-generated kernel Corollary
- On a simply connected domain, pathwise continuation glues to one holomorphic function Corollary
- ℝ/ℤ is not simply connected Corollary
- Universal covering spaces Definition
- Antipodal complements cover Sⁿ by simply connected sets with path-connected overlap for n≥2 Lemma
- Finite wedges of quotient circles have van Kampen covers at the wedge point Lemma
- A covering map induces an injective homomorphism on fundamental groups Theorem
- A space admitting a universal covering is semilocally simply connected Theorem
- Every nonempty convex subset of ℝⁿ is simply connected Theorem
- Every nonempty path-connected locally path-connected semilocally simply connected space has a universal cover Theorem
- Homological Serre spectral sequence Theorem
- Lie's second fundamental theorem Theorem
- Sⁿ is simply connected for every n≥2 Theorem
Dependency tree · two levels
18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- A. Hatcher, Algebraic Topology, Chapter 1, Proposition 1.6 (standard reference, not scraped)