clopen

Definitions

adjective mathematics Of a subset of a topological space, both open and closed.

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Since the intersection of all the clopen sets containing x with the closed set finite intersection property that there must be a finite intersection K 'of clopen sets containing x that is contained inside are clopen and do not contain K.

What's new 2009
Since the cylinder sets are clopen and generate the topology of of clopen sets generates the topology of X.

What's new 2009
The additional properties of being compact, Hausdorff, and totally disconnected are needed in order to recover the topology of X uniquely from the clopen algebra.

What's new 2009
Another application of the finite intersection property then reveals that every open neighbourhood of x contains at least one clopen set containing x, and so the clopen sets form a base as required.

What's new 2009
Show that two Stone spaces have isomorphic clopen algebras if and only if they are homeomorphic.

What's new 2009
Using compactness, we then conclude that every clopen set is the finite union of finite intersections of elements of is the power set of some set Y, then the Stone space given by Theorem 1 is the

What's new 2009
To state Stone's representation theorem we need another definition. of X to be the concrete Boolean algebra on X consisting of the clopen sets (i.e. sets that are both closed and open).

What's new 2009