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In mathematics, general topology (or point set topology) is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology , geometric topology , and algebraic topology .
The lattice of topologies on a set is a complemented lattice; that is, given a topology on there exists a topology ′ on such that the intersection ′ is the trivial topology and the topology generated by the union ′ is the discrete topology. [3] [4] If the set has at least three elements, the lattice of topologies on is not modular, [5 ...
General topology is the branch of topology dealing with the basic set-theoretic definitions and constructions used in topology. [11] [12] It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism , though usually most classify up to homotopy equivalence .
Cardinal functions are widely used in topology as a tool for describing various topological properties. [4] [5] Below are some examples.(Note: some authors, arguing that "there are no finite cardinal numbers in general topology", [6] prefer to define the cardinal functions listed below so that they never take on finite cardinal numbers as values; this requires modifying some of the definitions ...
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance.More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms ...
Absolutely closed See H-closed Accessible See . Accumulation point See limit point. Alexandrov topology The topology of a space X is an Alexandrov topology (or is finitely generated) if arbitrary intersections of open sets in X are open, or equivalently, if arbitrary unions of closed sets are closed, or, again equivalently, if the open sets are the upper sets of a poset.
A topological field of sets is called algebraic if and only if there is a base for its topology consisting of complexes. If a topological field of sets is both compact and algebraic then its topology is compact and its compact open sets are precisely the open complexes. Moreover, the open complexes form a base for the topology.