general topology and related areas of mathematics, the initial topology (or induced topology or strong topology or limit topology or projective topology) on...

In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators,...

In topology and related areas of mathematics, the set of all possible topologies on a given set forms a partially ordered set. This order relation can...

natural topology called the product topology. This topology differs from another, perhaps more natural-seeming, topology called the box topology, which...

is the initial topology, which for a given family of functions from a set X {\displaystyle X} into topological spaces is the coarsest topology on X {\displaystyle...

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...

coarsest topology may refer to: Initial topology, the most coarse topology in a certain category of topologies Trivial topology, the most coarse topology possible...

the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly...

a unique initial lift ( A → A i ) I {\displaystyle (A\to A_{i})_{I}} . In Top the initial lift is obtained by placing the initial topology on the source...

an existing topology, f is continuous with respect to this topology if and only if the existing topology is finer than the initial topology on S. Thus...

well-behaved with respect to initial topologies. Specifically, complete regularity is preserved by taking arbitrary initial topologies and the Tychonoff property...

topology. The simplest cases (which nevertheless cover many examples) are the initial topology and the final topology (Willard (1970)). The initial topology...

In topology, a subbase (or subbasis, prebase, prebasis) for a topological space X {\displaystyle X} with topology τ {\displaystyle \tau } is a subcollection...

generalized by the notion of a presheaf. Any topological space X has the initial topology induced by the family C ( X , S ) {\displaystyle C(X,S)} of continuous...

0. The uniform topology generated by the above uniformity is the initial topology induced by the family C(X). In general, this topology will be coarser...

constructed by placing the initial topology on the underlying set-theoretic inverse limit. This is known as the limit topology. The set of infinite strings...

areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric...

In topology, two points of a topological space X are topologically indistinguishable if they have exactly the same neighborhoods. That is, if x and y are...

certain sense. The discrete topology is the finest topology that can be given on a set. Every subset is open in the discrete topology so that in particular...

into X . {\displaystyle X.} Paths play an important role in the fields of topology and mathematical analysis. For example, a topological space for which there...

Lately, topology control algorithms have been divided into two subproblems: topology construction, in charge of the initial reduction, and topology maintenance...

normed spaces. They can be defined as topological vector spaces whose topology is generated by translations of balanced, absorbent, convex sets. Alternatively...

The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics....

rational numbers. The real numbers form an initial unbounded totally ordered set that is connected in the order topology (defined below). Ordered fields are...

= [0,1] to X such that f(0) = f(1). In other words, it is a path whose initial point is equal to its terminal point. A loop may also be seen as a continuous...

In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean...

triangulations established a new branch in topology, namely the piecewise-linear-topology (short PL-topology). Its main purpose is topological properties...

In topology, a coherent topology is a topology that is uniquely determined by a family of subspaces. Loosely speaking, a topological space is coherent...

In topology, the closure of a subset S of points in a topological space consists of all points in S together with all limit points of S. The closure of...

Algorithmic topology, or computational topology, is a subfield of topology with an overlap with areas of computer science, in particular, computational...