Examples
- The Euclidean plane R2 is simply connected, but R2 minus the origin (0,0) is not. If n > 2, then both Rn and Rn minus the origin are simply connected.
- Analogously: the n-dimensional sphere Sn is simply connected if and only if n ≥ 2.
- Every convex subset of Rn is simply connected.
- A torus, the (elliptic) cylinder, the Möbius strip, the projective plane and the Klein bottle are not simply connected.
- Every topological vector space is simply connected; this includes Banach spaces and Hilbert spaces.
- The special orthogonal group SO(n,R) is not simply connected for n ≥ 2; the special unitary group SU(n) is simply connected.
- The long line L is simply connected, but its compactification, the extended long line L* is not (since it is not even path connected).
- Similarly, the one-point compactification of R is not simply connected (even though R is simply connected).
Read more about this topic: Simply Connected Space
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