Definition:Either-Or Topology

Definition
Let $S = \closedint {-1} 1$ be the closed interval on the real number line from $-1$ to $1$.

Let $\tau \subseteq \powerset S$ be a subset of the power set of $S$ such that, for any $H \subseteq S$:
 * $H \in \tau \iff \paren {\set 0 \nsubseteq H \lor \openint {-1} 1 \subseteq H}$

where $\lor$ is the inclusive-or logical connective.

Then $\tau$ is the either-or topology, and $T = \struct {S, \tau}$ is the either-or space

That is, a set is open in $\tau$ if it does not contain $\set 0$ or it does contain $\openint {-1} 1$.

Also see

 * Either-Or Topology is Topology