Portal de Periódicos da CAPES

Locally connected spaces and their compactifications

1967; Duke University Press; Volume: 11; Issue: 3 Linguagem: Inglês

10.1215/ijm/1256054556

1945-6581

J. de Groot, R. H. McDowell,

Advanced Banach Space Theory

Example 5.3 seems to be of interest.Here we exhibit a subspace S of Euclidean three-space which is the union of a countable number of pairwise disjoint closed intervals, each nowhere dense in S, but S is nevertheless con- nected and locally connected.The authors wish to thank the referee for his many helpful suggestions and corrections, which improved the paper substantially.1. Components and quasicomponents 1.1.DEFINITIONS.Let p be a point in the space X.If the subset S of X is both open and closed in X, we say S is clopen in X.The component of p in X is the maximal connected subset of X containing p.