The Lower and Upper Limit Topologies on the Real Numbers

The Lower and Upper Limit Topologies on the Real Numbers

Recall from the Generating Topologies from a Collection of Subsets of a Set page that if $X$ is a set and $\mathcal B$ is a collection of subsets of $X$ then $\mathcal B$ is a basis for some topology $\tau$ if and only if the following two conditions are satisfed:

  • $X = \bigcup_{B \in \mathcal B} B$.
  • For all $B_1, B_2 \in \mathcal B$ and for all $x \in B_1 \cap B_2$ there exists a $B \in \mathcal B$ such that $x \in B \subseteq B_1 \cap B_2$.

If the collection $\mathcal B$ satisfies the two conditions above then the generated topology is:

(1)
\begin{align} \quad \tau = \left \{ U : U = \bigcup_{B \in \mathcal B^*} B, \mathcal B^* \subseteq \mathcal B \right \} = \left \{ \bigcup_{B \in \mathcal B^*} B : \mathcal B^* \subseteq \mathcal B \right \} \end{align}

We will now use this theorem to define two very important topologies on the set of real numbers $\mathbb{R}$.

Definition: The Lower Limit Topology on the set of real numbers $\mathbb{R}$, $\tau$ is the topology generated by all unions of intervals of the form $\{ [a, b) : a, b \in \mathbb{R}, a \leq b \}$.

Another name for the Lower Limit Topology is the Sorgenfrey Line.

Let's prove that $(\mathbb{R}, \tau)$ is indeed a topological space.

If $\tau$ is generated by unions of all intervals contained in $\mathcal B = \{ [a, b) : a, b \in \mathbb{R}, a \leq b \}$ then $\mathcal B$ is a basis for $\tau$.

We will show that the conditions from the Generating Topologies from a Collection of Subsets Satisfying Certain Conditions page are satisfied to verify that $\tau$ is indeed a topology.

Clearly the first condition is satisfied since:

(2)
\begin{align} \quad X = \bigcup_{a, b \in \mathbb{R}} [a, b) \end{align}

For the second condition, let $B_1 = [a, b), B_2 = [c, d) \in \mathcal B$. It is easier to visualize $B_1 \cap B_2$ with the following diagram:

Screen%20Shot%202015-09-24%20at%2010.54.13%20PM.png

In the first case above, we can easily find the interval $B = [a, b) = [c, d)$ such that for all $x \in B_1 \cap B_2$ we have that $x \in B \subseteq B_1 \cap B_2$.

For the second case above, we can also easily find $e, f \in \mathbb{R}$ such that $a < e < b$, $c < f < d$ such that $B = [e, f)$ and for which every $x \in B \subseteq B_1 \cap B_2$.

For the third and fourth cases above, we see that $B_1 \cap B_2 = \emptyset$.

Therefore $\tau$ is generated by $\mathcal B$ and is hence a topology on $\mathbb{R}$.

Of course, we can also define the upper limit topology on $\mathbb{R}$ analogously as follows:

Definition: The Upper Limit Topology on the set of real numbers $\mathbb{R}$, $\tau$ is the topology generated by all unions of intervals of the form $\{ (a, b] : a, b \in \mathbb{R}, a \leq b \}$.

In this case, $\tau$ is generated by $\mathcal B = \{ (a, b] : a, b \in \mathbb{R}, a \leq b \}$ and verifying that $\tau$ is indeed a topology follows similarly from above.

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