Exercise 23, § 4

Let f : X → Y be a continuous open mapping, X locally compact and Y Hausdorff.

a) If K is any compact connected subset of Y and C is any compact component of $\overline{f}(K)$, show that $f(C) = K$ (reduce to the case K = Y and use the Corollary to Proposition 6 of no. 4). Give an example of a non-compact component C' of $\overline{f}(K)$ such that $f(C') \neq K$.

b) Suppose in addition that Y is locally compact and locally connected. Show that if U is any connected open subset of Y and R is any relatively compact component of $\overline{f}(U)$ in X, then $f(R) = U$. [Observe that in a locally compact locally connected space, a connected open set U is the union of the compact connected sets contained in U and containing a given point $y_0 \in U$; then apply a).]

c) Suppose in addition that X is locally connected. Show that if K is any connected subset of Y which is either open or compact with a non-empty interior, and if H is any compact subset of X, then there is only a finite number of components of $\overline{f}(K)$ contained in H. [Use a) and b)].