In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the common meanings of open and closed are antonyms. But their mathematical definitions are not mutually exclusive. A set is closed if its complement is open, which leaves the possibility of an open set whose complement is itself also open, making both sets both open and closed, and therefore clopen.
== Examples ==
In any topological space X, the empty set and the whole space X are both clopen.
Now consider the space X which...
Well, the linear system which at least has one solution is called "consistent" linear system.
Find an equation involving g, h, and k that makes
this augmented matrix correspond to a consistent system:
$$\begin{bmatrix}
1 &-4 &7 &g \\
0 & 3 & -5 &h \\
-2 & 5 & -9 &k
\end{bmatrix}$$
Alright, ...
