nice topological space
nice category of spaces
convenient category of topological spaces
Freudenthal suspension theorem
CW-complex, Hausdorff space, second-countable space, sober space
compact space, paracompact space
connected space, locally connected space, contractible space, locally contractible space
topological vector space, Banach space, Hilbert space
point, real line, plane
sphere, ball, annulus
loop space, path space
Cantor space, Sierpinski space
long line, Warsaw circle
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A homotopy between continuous functions between topological spaces is called delayed if it starts out being constant near one boundary of the interval.
(If it is constant near both boundaries we say it has sitting instants).
For I=[0,1] the unit interval and X and Y any topological spaces, a continuous map F:X×I→Y is a delayed homotopy (between F(−,0) and F(−1)) if there exist t 0>0 such that F(x,t)=F(x,0) for all 0≤t≤t 0.
Delayed homotopies appear in an alternative characterization of Dold fibrations. See there for details.
If a continuous homotopy between two smooth functions is delayed at both ends of the inerval it may be approximated by a smooth homotopy . See Steenrod-Wockel approximation theorem.