Not really research level but here goes anyway: Suppose I have a topological space $X$ with a closed subset $K$ and a continuous map $f : X \times [0,1) \to X$ such that:

0) $f(x,0) = x$.

1) For all $k \in K, 0 \leq t < 1$, $f(k,t) \in K$.

2) For all $x \in X$, and all sequences $\{t_i\} \to 1$, there exists a subsequence $\{t_{i_j}\} \to 1$ and $k \in K$ such that $\{f(x,t_{i_j})\} \to k$.

Does it follows that $K$ is homotopy equivalent to, or a deformation retract of, $X$?