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For T2 spaces, it is shown that any continuous function from a connected locally connected space into a connected space with a dispersion point is a constant. The purpose of this note is to present a result of connected spaces with a dispersion point which seems to have gone unnoticed or is part of the folklore of mathematics, and then we would like to make a remark. There are many examples of connected spaces with a dispersion point, a classic one given by Knaster and Kuratowski [1] and for others, see [2], [3], [4]. For terminology, see [6]. All spaces are F2. Theorem. Any continuous function from a connected locally connected space into a connected space with a dispersion point is a constant. Proof. Suppose F is a connected space with a dispersion point/? and /is a continuous function from the locally connected space A into Y. Assume that/is not constant. This means that/(A) is a nondegenerate connected subset of Y and therefore contains p. By assumption /(A) — {p}j£0, and therefore the set G=f~1iY—{p}) is not empty and different from X. Denote by U any component of G. Since A is locally connected and G open, then U is open. We shall show that U is closed (this will contradict the assumption of connectedness of A). Since U is connected, so is/(£/), and since /(t7)<=/(G)<= Y— {/>}, we have p$fiU). It follows that/(i7) is a single point q¥^pNow to show that U is closed, let xeD. Hence /(x)eCl(/(i/)). But, since fiU) = {q}, we have Cl(/(i7)) = {^} and hence fix)=q¿¿p, i.e. xef-\Y-{p})=G. Denote by W the component of G which contains x. Thus xeU(~\W and it follows that W= U since W and U are open sets, either disjoint or identical. Hence xell. Thus U is closed. Received by the editors November 24, 1970. AMS 1970 subject classifications. Primary 54C05, 54D05.

Continuous functions from a connected locally connected space into a connected space with a dispersion point