Connectedness Modulo Realcompactness

In this article, we study connectedness modulo P when P is realcompactness. We show that a Tychonoff space X is connected modulo realcompactness if and only if clβX (υX\X) is connected. We state and prove several results dual to the standard existing results about connected spaces. We show that connected modulo realcompactness spaces are preserved under perfect open continuous surjections. The concept of “connectedness modulo an ideal” was presented by Koushesh and studied when the ideal is RX generated by the set of all subsets of X with realcompact closure for normal spaces. As another main subject in this paper, we study connectedness modulo the ideal RX generally for completely regular Hausdorff spaces. Especially, we show that whenever f : X → Y is a perfect open continuous surjection and X is connected modulo the ideal RX , then Y is connected modulo the ideal RY .