Sorgenfrey Line is Hausdorff

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Theorem

Let $T = \left({\R, \tau}\right)$ be the Sorgenfrey line.

Then $T$ is Hausdorff.


Proof

Take $x, y \in \R$ such that $x \ne y$.

WLOG, assume that $x < y$.

From Real Numbers are Close Packed:

$\exists t \in \R: x < t < y$

Then:

$\left[{x \,.\,.\, t}\right) \cap \left[{t \,.\,.\, y+1}\right) = \varnothing$

We also have that $x \in \left[{x \,.\,.\, t}\right)$ by definition of half-open interval.

Also, as $t < y < y+1$ it is clear that $y \in \left[{t \,.\,.\, y+1}\right)$.

By definition of the Sorgenfrey line, both are open in $T$.

Thus we have found two disjoint subsets of $\R$ which are open in $T$, such that one contains $x$ and the other contains $y$.

Hence the Sorgenfrey Line is Hausdorff by definition.

$\blacksquare$


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