# Article

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Keywords:
insertion; $\sigma$-topology; $\sigma$-ring; perfectness; normality; upper measurable function; lower measurable function; measurable function
Summary:
A family of subsets of a set is called a $\sigma$-topology if it is closed under arbitrary countable unions and arbitrary finite intersections. A $\sigma$-topology is perfect if any its member (open set) is a countable union of complements of open sets. In this paper perfect $\sigma$-topologies are characterized in terms of inserting lower and upper measurable functions. This improves upon and extends a similar result concerning perfect topologies. Combining this characterization with a $\sigma$-topological version of Katětov-Tong insertion theorem yields a Michael insertion theorem for normal and perfect $\sigma$-topological spaces.
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