On Compactifications with Path Connected Remainders

We prove that every separable and metrizable space admits a metrizable compactification with a remainder that is both path connected and locally path connected. This result answers a question of P. Simon. Connectedness and compactness are two fundamental topological properties. A natural question is whether a given space admits a connected (Hausdorff) compactification. This question has been studied extensively; see [1, 5, 7, 9, 11]. We mention both a positive and a negative result. It was proved by Alas, Tkačenko, Tkachuk, and Wilson [1] that a separable and metrizable space admits a metrizable and connected compactification provided the space contains no proper subsets that are open and compact. Confirming a conjecture of van Douwen [4], Emeryk and Kulpa [5] proved that the famed Sorgenfrey line does not admit a connected compactification. We focus on the question whether a given space X admits a compactification C such that the remainder C \ X is connected. Our research was prompted by a question asked by Simon [10], namely whether the space one obtains when one removes a countable dense subset from the unit square [0, 1] has some compactification such that the remainder is connected. Simon conjectured that the answer should be no. We present a general result that in particular shows that the answer to Simon’s question is yes. Theorem 1. Every separable and metrizable space admits a metrizable compactification with a remainder that is both path connected and locally path connected. Proof. Let X be a separable and metrizable space. Select a metrizable compactification C of X and let d be a metric on C that generates the topology. Select a countable dense subset A in the remainder C \ X. If C \X is finite, then X is locally compact and we can use the one-point compactification, so we may assume that A is infinite. Let {ai : i ∈ N} enumerate A in such a way that ai = aj if i = j. The idea of the proof is as follows. We first construct a new compactification D of X from C by “blowing up” the points an to intervals In. We then “glue” the In together, producing a quotient space E of D that also compactifies X and in which the image of ⋃∞ n=1 In is a path connected set that is dense in the remainder. Received by the editors October 30, 2007. 2000 Mathematics Subject Classification. Primary 54D40, 54D05.