12 Facts About General topology

In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology.

General topology grew out of a number of areas, most importantly the following:.

Base 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 are useful because many properties of topologies can be reduced to statements about a base that generates that topology—and because many topologies are most easily defined in terms of a base that generates them.

The set of all open intervals forms a base or basis for the General topology, meaning that every open set is a union of some collection of sets from the base.

Possible topologies on a fixed set X are partially ordered: a General topology t1 is said to be coarser than another General topology t2 if every open subset with respect to t1 is open with respect to t2.

Product General topology on X is the General topology generated by sets of the form pi, where i is in I and U is an open subset of Xi.

In other words, the sets {pi} form a subbase for the topology on X A subset of X is open if and only if it is a union of intersections of finitely many sets of the form pi.

Pointless General topology is an approach to General topology that avoids mentioning points.

The ideas of pointless General topology are closely related to mereotopologies, in which regions are treated as foundational without explicit reference to underlying point sets.

Dimension theory is a branch of general topology dealing with dimensional invariants of topological spaces.

In General topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space.

Set-theoretic topology is a subject that combines set theory and general topology.