A completely regular space which is the $T_1$-complement of itself

A Pseudocompact Completely Regular Frame which is not Spatial

Compact regular frames are always spatial. In this note we present a method for constructing non-spatial frames. As an application we show that there is a countably compact (and hence pseudocompact) completely regular frame which is not spatial.

A metrizable completely regular ordered space

We construct a completely regular ordered space (X, T ,≤) such that X is an I-space, the topology T of X is metrizable and the bitopological space (X, T ♯,T ♭) is pairwise regular, but not pairwise completely regular. (Here T ♯ denotes the upper topology and T ♭ the lower topology of X.)

The Complement of a D-Tree is Pure Shellable

Compactification of a Map Which Is Mapped to Itself

We prove that if T : X → X is a selfmap of a set X such that ⋂ {T nX : n ∈ N} is a one-point set, then the set X can be endowed with a compact Hausdorff topology so that T is continuous.

Ompactification of Completely Regular Frames based on their Cozero Part

 Let L  be a frame. We denoted the set of all regular ideals of cozL by rId(cozL) . The aim of this paper is to study these ideals. For a  frame L , we show that  rId(cozL) is a compact completely regular frame and the map jc : rId(cozL)→L  given by jc (I)=⋁I   is a compactification of L which is isomorphism to its  Stone–Čech compactification and is proved that jc have a right adjoint rc : L →...