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...

closed 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. A set can...

and are called clopen sets. Given a topological space ( X , τ ) {\displaystyle (X,\tau )} , the following statements are equivalent: a set A ⊆ X {\displaystyle...

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. A topological...

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...

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...

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...

poset. Almost discrete A space is almost discrete if every open set is closed (hence clopen). The almost discrete spaces are precisely the finitely generated...

non-empty open sets. The only subsets of X {\displaystyle X} which are both open and closed (clopen sets) are X {\displaystyle X} and the empty set. The only...

all spaces whose base consists of clopen sets.) The above definitions of open and closed sets provide the first two sets Σ 1 0 {\displaystyle \mathbf {\Sigma...

states that every Boolean algebra is isomorphic to the Boolean algebra of clopen sets of the Stone space S ( B ) {\displaystyle S(B)} ; and furthermore, every...

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...

a pair of disjoint non-empty open sets. Equivalently, a space is connected if the only clopen sets are the empty set and itself. Locally connected. A space...

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...

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...

(that is, a clopen set). Consider the real line R {\displaystyle \mathbb {R} } with the usual topology (that is, the topology whose basis sets are open intervals)...

to the algebra of clopen subsets of its Stone space S(B). The isomorphism sends an element b ∈ B {\displaystyle b\in B} to the set of all ultrafilters...

bases consisting of clopen sets are homeomorphic to each other. The topological property of having a base consisting of clopen sets is sometimes known...

\{p\}} is totally separated (for each two points x and y there exists a clopen set containing x and not containing y) then p is an explosion point. A space...

mathematics, a Borel set is any subset of a topological space that can be formed from its open sets (or, equivalently, from closed sets) through the operations...

regular open set is an open set and every regular closed set is a closed set. Each clopen subset of X {\displaystyle X} (which includes ∅ {\displaystyle \varnothing...

set of this refinement. A topological space is zero-dimensional with respect to the small inductive dimension if it has a base consisting of clopen sets...

with respect to the small inductive dimension has a base consisting of clopen sets. Every such space is regular. As described above, any completely regular...

causet, from causal and set cermet, from ceramic and metal chemokine, from chemotactic and cytokine clopen set, from closed-open set contrail, from condensation...

it contains a path-connected neighbourhood of {e}; and therefore is a clopen set. The quotient group G/G0 is called the group of components or component...

Topological space Topological property Open set, closed set Clopen set Closure Boundary Density G-delta set, F-sigma set Closeness Neighborhood Continuity (topology)...

semantics for S5. Hence S5-algebra is a synonym for monadic Boolean algebra. Clopen set Cylindric algebra Interior algebra Kuratowski closure axioms Łukasiewicz–Moisil...

Open set Clopen set Fσ set Gδ set Compact set Relatively compact set Regular open set, regular closed set Connected set Perfect set Meagre set Nowhere...

negation map, ultrafilters) are used to obtain the set of points of a topological space, whose clopen sets are isomorphic to the original Boolean algebra...

definitions worth mentioning are:. Stone (1936) A Boolean algebra is the set of all clopen sets of a topological space. It is no limitation to require the space...