In topology and related areas of mathematics, the final topology on a setX is the strongest topology to make a family of functions into Xcontinuous. Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces. ... History Main article: History of mathematics In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. ... The notion of a set is one of the most important and fundamental concepts in modern mathematics. ... In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ...
Given a set X and a familiy of topological spaces(Yi,τi) with functions Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...
the final topologyτ on X is the strongest topology such that each
The direct sum topology is the final topology with respect to the family of canonical injections.
For quotient spaces in linear algebra, see quotient space (linear algebra). ... For quotient spaces in linear algebra, see quotient space (linear algebra). ...
Properties
A subset of X is open (closed) if and only if it is open (closed) in all Yi.
A function g from X to some space Z is continuous if and only if for each fi is continuous.
In mathematics, philosophy, logic and technical fields that depend on them, iff is used as an abbreviation for if and only if. It is often, not always, written italicized: iff. ...
In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense.
In topology and related areas of mathematics, the initial topology (projective topology or weak topology) on a set, with respect to a family of functions on, is the coarsest topology on X which makes those functions continuous.
In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology (or the relative topology, or the induced topology).
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