Hausdorffness is hereditary
This article gives the statement, and possibly proof, of a topological space property (i.e., Hausdorff space) satisfying a topological space metaproperty (i.e., subspace-hereditary property of topological spaces)
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Further information: Hausdorff space
A topological space is Hausdorff if given distinct points there exist disjoint open subsets containing respectively.
Further information: subspace topology
Given: A topological space , a subset of . Two distinct points .
To prove: There exist disjoint open subsets of such that .
|Step no.||Assertion/construction||Facts used||Given data used||Previous steps used||Explanation|
|1||are distinct points of|| are distinct points of
|2||There exist disjoint open subsets of such that .||is Hausdorff||Step (1)||Step-given direct|
|3||Define and .|
|4||are open subsets of .||definition of subspace topology||Steps (2), (3)||By Step (2), are open, so by the definition of subspace topology, are open as per their definitions in Step (3).|
|5||are disjoint.||Steps (2), (3)||follows directly from being disjoint|
|6||Steps (2), (3)||By Step (3), . By Step (2), , and we are also given that , so . Similarly, .|
|7||are the desired open subsets of .||Steps (4)-(6)||Step-combination direct, it's what we want to prove.|
This proof uses a tabular format for presentation. Learn more about tabular proof formats|View all pages on facts with proofs in tabular format
- Topology (2nd edition) by James R. Munkres, More info, Page 100, Theorem 17.11, Page 101, Exercise 12 and Page 196 (Theorem 31.2 (a))