ON TOPOLOGICAL STRUCTURE OF SPACES THAT ARE MILDLY CONNECTED-CLEAVABLE OVER THE REALS

Abstract

We prove the following theorem (8): If X is a nowhere hereditarily disconnected homogeneous space metrizable by a complete metric, and X is cleavable over R along every punctured closed connected subset, then X is locally connected. Using this result, we establish the next theorem (Theorem 15): Suppose that X is an infinite homogeneous connected locally compact metrizable space. Suppose also that X is cleavable over R along every punctured closed connected subset. Then X is homeomorphic to the space R of real numbers.