Two questions about connected compactifications by Petr Simon

Topology Atlas Document # qaaa-22

Two questions about connected compactifications

Petr Simon

Question. Consider the square of the closed unit interval with all points with both coordinates rational removed, i.e., IxI \ QxQ. Has this space a compactification with connected remainder?

The answer should be no.

Question. Has the square of the Sorgenfrey line some connected compactification?

Emeryk and Kulpa years ago answered a question due to Eric van Douwen, namely, they proved that the Sorgenfrey line has no connected compactification, In fact, they showed a bit more: the Sorgenfrey line cannot be densely embedded into any regular connected space, and in the same paper they found a Hausdorff connected space, containing the Sorgenfrey line densely.

The two spaces from my questions are perhaps the easiest cases, where Emeryk-Kulpa's method must fail, so something different is needed.

References

[1] A. Emeryk and W. Kulpa, The Sorgenfrey line has no connected compactification, Comment. Math. Univ. Carolinae 18 (1977), no. 3, 483-487; MR57:1422

Received: February 24, 2003