Some Remarks on Continuous Transformations

Suppose / is a single-valued transformation of the metric space M onto the metric space fM. Many of the theorems of topology assert that if / is continuous, then it must have various other properties also. We deal here with the problem of determining under what circumstances certain of these properties are actually equivalent to continuity. The properties considered are: (1) fX is compact for every compact XQM; (2) fY is connected for every connected YQM; (3) f~lq is closed for every point qGfM; (4) f~l{z:zE:fM and p(z, q) =e} is closed for each e>0 and q(EJM. By an "i-map" on M we will mean a transformation / satisfying the condition (i), where * = 1, 2, 3, or 4. A basic lemma is (L) Suppose f is a l-map on M, discontinuous at the point pGM. Then there are a point q(EJM and a sequence pa oS points oS M such that pa—>p, q^SPi andSPi = qS°r each i. Proof. Clearly there are an open set V3SP and a sequence xa of distinct points of M such that xa-^>p and /xi = g,i^F. If the desired conclusion fails to hold, then each g< is the image of only finitely many x/s, and there is an infinite subsequence ya of xa such that S= {yi, y2, ■ ■ ■ } maps biuniquely under/. But SKJ\p] is compact, so/SW {Sp} is compact, and hence, since/S misses V,SS is compact. Thus for some j, SVi is an accumulation point of /S. But (S— {yj}) ^J{p} is compact and hence (as above) /S — {/jy} is compact, a contradiction completing the proof. Since/-1g in (L) is clearly not closed, we have (A) Every 1, 3-map on M is continuous. A corollary is the fact that every biunique l-map is continuous. Our next result is (B) IS M is locally connected at p, then every 1, 2-map on M is continuous at p. IS M is not locally connected at p, then M admits a realvalued 1, 2-map which is discontinuous at p. Proof. To prove the first assertion, suppose M is locally connected at p but admits a 1, 2-map/ (onto some metric space) which is discontinuous at p. Let pa and q be as in (L). By local connectedness of M at p, there are a subsequence xa of pa and a sequence Ca of con-