On function spaces related to d-spaces

Abstract

For certain property ℜ of topological <math><msub is="true"><mrow is="true"><mi is="true">T</mi></mrow><mrow is="true"><mn is="true">0</mn></mrow></msub></math>-spaces, we prove that if <math><msub is="true"><mrow is="true"><mi is="true">T</mi></mrow><mrow is="true"><mn is="true">0</mn></mrow></msub></math>-space <math><mi mathvariant="double-struck" is="true">Y</mi></math> has property ℜ, then <math><mo stretchy="false" is="true">[</mo><mi mathvariant="double-struck" is="true">X</mi><mo is="true">,</mo><mi mathvariant="double-struck" is="true">Y</mi><mo stretchy="false" is="true">]</mo></math> endowed with the Isbell topology possesses ℜ for any <math><msub is="true"><mrow is="true"><mi is="true">T</mi></mrow><mrow is="true"><mn is="true">0</mn></mrow></msub></math>-space <math><mi mathvariant="double-struck" is="true">X</mi></math> ; and if <math><mi mathvariant="double-struck" is="true">X</mi></math> has property ℜ for some non-empty space <math><mi mathvariant="double-struck" is="true">X</mi></math>, then so does <math><mi mathvariant="double-struck" is="true">Y</mi></math>.