Characterizing Continuity by Preserving Compactness and Connectedness

Let us call a function f from a space X into a space Y preserving if the image of every compact subspace of X is compact in Y and the image of every connected subspace of X is connected in Y . By elementary theorems a continuous function is always preserving. Evelyn R. McMillan [6] proved in 1970 that if X is Hausdorff, locally connected and Frèchet, Y is Hausdorff, then the converse is also true: any preserving function f : X → Y is continuous. The main result of this paper is that if X is any product of connected linearly ordered spaces (e.g. if X = R) and f : X → Y is a preserving function into a regular space Y , then f is continuous. Let us call a function f from a space X into a space Y preserving if the image of every compact subspace of X is compact in Y and the image of every connected subspace of X is connected in Y . By elementary theorems a continuous function is always preserving. Quite a few authors noticed— mostly independently from each other—that the converse is also true for real functions: a preserving function f : R → R is continuous. (The first paper we know of is [9] from 1926!) Whyburn proved [13] that a preserving function from a space X into a Hausdorff space is always continuous at a first countability and local connectivity point of X. Evelyn R. McMillan [6] proved in 1970 that if X is Hausdorff, locally connected and Frèchet, moreover Y is Hausdorff, then any preserving function f : X → Y is continuous. This is quite a significant and deep result that is surprisingly little known. 2000 Mathematics Subject Classification. 54C05, 54D05, 54F05, 54B10.