Hausdorffness is hereditary

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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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Statement

Any subspace of a Hausdorff space is Hausdorff in the subspace topology.

Definitions used

Hausdorff space

Further information: Hausdorff space

A topological space X is Hausdorff if given distinct points a,b \in X there exist disjoint open subsets U,V containing a,b respectively.

Subspace topology

Further information: subspace topology

If A is a subset of X, we declare a subset V of A to be open in A if V = U \cap A for an open subset U of X.

Proof

Given: A topological space X, a subset A of X. Two distinct points x_1,x_2 \in A.

To prove: There exist disjoint open subsets U_1,U_2 of A such that x_1 \in U_1,x_2 \in U_2.

Proof:

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 x_1, x_2 are distinct points of X x_1,x_2 are distinct points of A
A \subseteq X
2 There exist disjoint open subsets V_1, V_2 of X such that x_1 \in V_1, x_2 \in V_2. X is Hausdorff Step (1) Step-given direct
3 Define U_1 = V_1 \cap A and U_2 = V_2 \cap A.
4 U_1, U_2 are open subsets of A. definition of subspace topology Steps (2), (3) By Step (2), V_1,V_2 are open, so by the definition of subspace topology, U_1, U_2 are open as per their definitions in Step (3).
5 U_1, U_2 are disjoint. Steps (2), (3) follows directly from V_1,V_2 being disjoint
6 x_1 \in U_1, x_2 \in U_2 x_1,x_2 \in A Steps (2), (3) By Step (3), U_1 = V_1 \cap A. By Step (2), x_1 \in V_1, and we are also given that x_1 \in A, so x_1 \in V_1 \cap A = U_1. Similarly, x_2 \in U_2.
7 U_1,U_2 are the desired open subsets of A. 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

References

Textbook references

  • 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))