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...

set. A set may be both open and closed (a clopen set). The empty set and the full space are examples of sets that are both open and closed. Open sets...

closed and are called clopen sets. The ray [ 1 , + ∞ ) {\displaystyle [1,+\infty )} is closed. The Cantor set is an unusual closed set in the sense that it...

logics. Given a topological space the clopen sets trivially form a topological field of sets as each clopen set is its own interior and closure. The Stone...

each other, the empty set is also closed, making it a clopen set. Moreover, the empty set is compact by the fact that every finite set is compact. The closure...

the relative topology on the Cantor set, the points have been separated by a clopen set. Consequently, the Cantor set is totally disconnected. As a compact...

Cylinder sets are clopen sets. As elements of the topology, cylinder sets are by definition open sets. The complement of an open set is a closed set, but...

complement is open). A subset of X may be open, closed, both (clopen set), or neither. The empty set and X itself are always both closed and open. A base (or...

is locally connected, then, as above, C x {\displaystyle C_{x}} is a clopen set containing x, so Q C x ⊆ C x {\displaystyle QC_{x}\subseteq C_{x}} and...

complement is open). A subset of X may be open, closed, both (a clopen set), or neither. The empty set and X itself are always both closed and open. An open subset...