# 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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## 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))