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In mathematics, general topology or point set topology is that branch of topology which studies elementary properties of topological spaces and structures defined on them. It grew out of a number of areas, such as the detailed study of sets of points (as subsets of the real line, understood), the manifold concept, the metric spaces and the early days of functional analysis. It was codified, in much its form for the remainder of the twentieth century, around 1940. It captures, one might say, almost everything in the intuition of continuity, in a technically adequate form that can be applied in every area of mathematics.More specifically, it is in general topology that basic notions, such as:
- open and closed sets;
- compact spaces;
- continuous functions;
- convergence of sequenceThis is a page about mathematics. For other usages of "sequence", see: sequence (non-mathematical). In mathematics, a sequence is a list of objects (or events) which have been ordered in a sequential fashion; such that each member either comes before, ors, netTopology In mathematics the term net has at least two meanings. See the glossary of Riemannian and metric geometry for its meaning for metric spaces. This article is about its meaning in topology, where the concept of a net is a generalization of that ofs, and filterIn mathematics, a filter is a special subset of a partially ordered set. A frequently used special case is the situation that the ordered set under consideration is just the power set of some set, ordered by set inclusion. Filters appear in order and latts;
- separation axiomIn topology and related fields of mathematics, there are several restrictions that one often makes on the kinds of topological spaces that one wishes to consider. Some of these restrictions are given by the separation axioms . These are sometimes called Ts:
are defined and theorems about them are proved.
Other more advanced notions also appear, but are usually related directly to these fundamental concepts, without reference to other branches of mathematics.
Other main branches of topology are algebraic topologyTopology Algebraic topology Abstract algebra Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. The method of algebraic invariants The goal is to take topological spaces, and further ca, geometric topologyTopology Geometric topology In mathematics, geometric topology is the study of manifolds and their embeddings, with representative topics being knot theory and braid groups. It has come over time to be almost synonymous with low-dimensional topology conce, and differential topology. As the name implies, general topology provides the common foundation for these areas.
See glossary of general topology for detailed definitions; and the list of general topology topics.
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