In this paper, we investigate the relations between the stratifiable structure of spaces and the insertion of semi-continuous functions and give some characterizations of perfect spaces, semi-stratifiable spaces and K-semi-stratifiable spaces.

van Mill et al. posed in 'Classes defined by stars and neighborhood assignments', Topology Appl. 154 (2007), 2127-2134 the following question: Is a star-compact space metrizable if it has a G δ -diagonal? In this paper, we give a negative answer to this question.2000 Mathematics subject classification: 54D20, 54E35.

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