- 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...
-
closed are
called clopen. 0 and 1 are
clopen. An
interior algebra is
called Boolean if all its
elements are open (and
hence clopen).
Boolean interior...
- and b {\displaystyle b} , the
interval [ a , b ) {\displaystyle [a,b)} is
clopen in R l {\displaystyle \mathbb {R} _{l}} (i.e., both open and closed). Furthermore...
-
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 is locally...
-
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...
- ≰ 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...
- in the real numbers. Some sets are both open and
closed and are
called clopen sets. The ray [ 1 , + ∞ ) {\displaystyle [1,+\infty )} is closed. The Cantor...
-
intermediate 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...
