Abstrakt
We are analyzing some of H. Simmons, S.B.
Niefield and K. I.
Rosenthal results concerning sublocales induced by subspaces. H.
Simmons was concerned with the question when the coframe of sublocales is Boolean; he recognized the role of the axiom T_D for the relation of certain degrees of scatteredness but did not emphasize its role in the relation between sublocales and subspaces. S.B.
Niefield and K. I.
Rosenthal just mention this axiom in a remark about Simmons' result. In this paper we show that the role of TD in this question is crucial.
Concentration on the properties of TD-spaces and technique of sublocales in this context allows us to present a simple, transparent and choice-free proof of the scatteredness theorem.