Hausdorffness is hereditary

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

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
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:

Textbook references

 * , Page 100, Theorem 17.11, Page 101, Exercise 12 and Page 196 (Theorem 31.2 (a))