Monotone normality is hereditary
This article gives the statement, and possibly proof, of a topological space property (i.e., monotonically normal space) satisfying a topological space metaproperty (i.e., subspace-hereditary property of topological spaces)
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Suppose is a monotonically normal space with a monotone normality operator . Suppose is a subspace of . Then, is also a monotonically normal space, with a monotone normality operator defined in terms of .
- Monotonically normal implies collectionwise normal
- Monotonically normal implies hereditarily collectionwise normal
- Monotonically normal implies hereditarily normal
Given: A monotonically normal space with monotone normality operator . A subspace of .
To prove: We can define a monotone normality operator on in terms of .
Proof: Note that the part about being implying being is immediate.
Suppose and are disjoint closed subsets of . Then, by the definition of subspace topology, we have that is empty and is empty.
We define the monotone normality operator on as follows:
We claim the following:
- is open in : Note first that since is , the point is closed. Also, is closed, and disjoint from . Thus, is open for each . Thus, the union of all of these is open in , and so, by the definition of subspace topology, the intersection with is open in .
- The closure of in is disjoint from : Consider for . This is an open subset of by definition, and for all . Thus, we obtain that . The last set is closed, and does not contain any of the elements , so is disjoint from . Thus, is contained in a closed subset of that is disjoint from . Intersecting with , we obtain that is contained in a closed subset of that is disjoint from . Thus, the closure of in is disjoint from .
- If and , with all closed in , , then : Since , we have , so for all . Thus, . In turn, it is clear that . Thus, .