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(topology)

pointless topology

noun

(mathematics) An approach to topology that avoids mentioning points.

The study of pointless topology is fascinating.

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initial topology

noun

(topology) Given a set X, a family of topological spaces (M_i,T_i)_(i∈I) and a family of functions f_i:X→M_i, the initial topology is the coarsest topology on X such that all f_i are continuous.

The initial topology on a set X is the coarsest topology on X that makes all the functions in a given collection of functions from X to topological spaces continuous.

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final topology

noun

(topology) Given a set X, a family of topological spaces (M_i,T_i)_(i∈I) and a family of functions f_i:M_i→X, the final topology is the finest topology on X such that all f_i are continuous.

The final topology is determined by the collection of open sets in X.

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point-set topology

noun

The general field of topology, not restricting attention to specific classes of spaces, and not using algebraic topology.

Point-set topology is a branch of mathematics that deals with the general field of topology.

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disjoint union topology

noun

(topology) A topology which is applicable to the disjoint union of a given set of topological spaces and is the largest (most inclusive) topology that preserves the continuity of each contributing space as represented in the union.

The disjoint union topology is a way to combine two topological spaces.

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topological space

noun

(topology, formally) An ordered pair (X, τ), where X is a set and τ, called the topology, is a collection of subsets of X which satisfies certain axioms and whose elements are called the open sets (or alternatively, for a different set of axioms, the closed sets);