Totally nonconnected im kleinen continua

Abstract

If a connected topological space T' is not connected im kleinen at the point p of T', then there is some open set D' containing p such that p is a boundary point of the p-component of D'. This paper shows that much stronger conditions than those above hold at some points of a Baire topological continuum' which is not connected im kleinen at any point of a certain domain intersection subset. Limits to results of this kind are shown by an example of a bounded plane continuum that is connected im kleinen at each point of a dense inner limiting (i.e., Ga) subset but is not locally connected at any point. DEFINITION. A topological continuum T' is totally nonconnected im kleinen on a subset A of T' if T' is not connected im kleinen at any point of A (i.e., if each point p of A is contained in some open subset U of T' such that p is a boundary point, relative to T', of the pcomponent of U). DEFINITION. If (1) T' is a topological continuum, (2) Z' is the least cardinal number of any topological basis for T', (3) the set U is an open subset of T', and (4) the subset A of U is the nonvacuous common part of not more than Z' open subsets of T', each dense in U, then A is a dense-domain intersection subset of U (relative to T'). If, in addition to (1), (2) and (3), U is not the sum of Z', or fewer than Z', closed nowhere dense subsets of T', then T' is Baire topological on U. If T' is Baire topological on each open subset of U, then T' is locally Baire topological on U.