We consider poset models of topological spaces and show that every T1-space has an bounded complete algebraic poset model, thus give a positive answer to a question asked in a recent paper by Waszkiewicz. It is also proved that every T1-space is homeomorphic to the maximal point space of a d-space. In the classic general topology, people are mainly interested in the spaces which satisfy at leas...

Building on [4], [8] and [9] we study which cardinals are characterizable by a Scott sentence, in the sense that φM characterizes κ, if it has a model of size κ, but not of κ. We show that if אα is characterizable by a Scott sentence and β < ω1, then אα+β is characterizable by a Scott sentence. If 0 < γ < ω1, then the same is true for 2אα+γ . Also, אא0 α is characterizable by a Scott sentence. ...

in this paper, the definition of meet-continuity on $l$-directedcomplete posets (for short, $l$-dcpos) is introduced. as ageneralization of meet-continuity on crisp dcpos, meet-continuity on$l$-dcpos, based on the generalized scott topology, ischaracterized. in particular, it is shown that every continuous$l$-dcpo is meet-continuous and $l$-continuous retracts ofmeet-continuous $l$-dcpos are al...