A semiregular space is a topological space whose regular open sets form a base. Compare to the stronger elementary properties of a regular space. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... This is a glossary of some terms used in the branch of mathematics known as topology. ... In mathematics, a base (or basis) B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T. Bases... In topology and related fields of mathematics, regular spaces and T3 spaces are particularly nice kinds of topological spaces. ...
X is a regular space iff, given any closed set F and any point x that does not belong to F, there are a neighbourhood U of x and a neighbourhood V of F that are disjoint.
Most topological spaces studied in mathematical analysis are regular; in fact, they are usually completely regular, which is a stronger condition.
On the other hand, spaces that are regular but not completely regular, or preregular but not regular, are usually constructed only to provide counterexamples to conjectures, showing the boundaries of possible theorems.
Feeds |
Contact