- 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...
- {\displaystyle X}
itself are
always clopen.
These two sets are the most well-known
examples of
clopen subsets and they show that
clopen subsets exist in
every topological...
- x\},}
where b is an
element of B.
These sets are also
closed and so are
clopen (both
closed and open). This is the
topology of
pointwise convergence of...
- with
closed manifold. Sets that are both open and
closed and are
called clopen sets.
Given a
topological space ( X , τ ) {\displaystyle (X,\tau )} , the...
-
connected components of a
locally connected space are also open, and thus are
clopen sets. It
follows that a
locally connected space X is a
topological disjoint...
- 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
subsets of X {\displaystyle...
- \{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...
- ≰ y {\displaystyle \scriptstyle x\,\not \leq \,y} , then
there exists a
clopen up-set U of X such that x∈U and y∉ U. (This
condition is
known as the Priestley...
- the same as Δ0 1) Σ0 0 = Π0 0 = Δ0 0 (if defined) Δ0 1 =
recursive Δ0 1 =
clopen Σ0 1 =
recursively enumerable Π0 1 = co-recursively
enumerable Σ0 1 = G...
-
respect to the
small inductive dimension if it has a base
consisting of
clopen sets. The
three notions above agree for separable,
metrisable spaces.[citation...
