On a certain class of set theoretic properties
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One of the most useful results in introductory topology is the theorem that ifS is connected andC is a connected subset such thatS −C =X∪Y, separate («∪ » means union), thenX∪C andY∪C are connected. In this paper a study is made of other set properties satisfying a similar theorem. That is, ifP ⊂ S both have a property & andS −P=X∪Y separate, thenP∪X andP∪Y have this property. Furthermore, many properties have corresponding localizations, for example local connectedness, and it is determined under what conditions the corresponding local property satisfies the above theorem if the original property does.