Answer to Question #217686 in Differential Geometry | Topology for Prathibha Rose

Question #217686
Prove that the property being a Hausdroff space is a topological and hereditary property
1
Expert's answer
2021-07-22T18:00:13-0400

"\\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:}\\\\\n(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. }"


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