CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
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 $F: X\times I\to Y$ is a delayed homotopy (between $F(-,0)$ and $F(-1))$ if there exist $t_0\gt 0$ such that $F(x,t)=F(x,0)$ for all $0\leq t\leq t_0$.
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.