Question

Prove that the property being a Hausdroff space is a topological and hereditary property

Answer 1

$\text{Let (X,$\tau$) be a Hausdorff topological space and let x,y $\in X.$ Since $A \subseteq X.$ So since }\\\text{x is Hausdroff there exists open neighbourhood u of X and v of Y such that }\\u\cap v=\empty\\\text{Since u and v contains x and y. We have that $A \cap u$ and $A \cap v$ are open neighbourhoods}\\\text{of X and Y in A. Moreover we see that:}\\ (A \cap U) \cap(A \cap C)=A\cap(u\cap v)=A \cap \empty=\empty\\\text{So for all x,y $\in A$, there exists open neighbourhoods so that A $\cap u$ and $A \cap v$= $\empty$.}\\\text{So (A, $\tau$) is Hausdroff. This shows that the Hausdroff property is hereditary. }$