# Houston Problem Book Problem 42

**Problem 42 (HPB):** Suppose $X$ is a compact metric space and $S$ is the set of all stable points of $X$. Suppose $\mathrm{dim} \hspace{2pt} X$ is finite and $a$ is an infinite cycle in $S$. Is it true that if $a \sim 0$ in $X$, then $a \sim 0$ in $S$? (Asked by T. Ganea, 7 March 1973)

**Answer:** Yes, solved by Namioka. (W. Kuperberg and P. Minc 28 March 1979)