\begin{definition}[Definition:Either-Or Topology]
Let $S = \closedint {-1} 1$ be the closed interval on the real number line from $-1$ to $1$.
Let $\tau = \set {U \in \powerset S: \paren {\set 0 \nsubseteq U} \lor \paren {\openint {-1} 1 \subseteq U} }$ where:
:$\powerset S$ is the power set of $S$
:$\lor$ 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 $\tau$ {{iff}} it ''does not'' contain $\set 0$ or it ''does'' contain $\openint {-1} 1$.
\end{definition}