mathematics, the particular point topology (or included point topology) is a topology where a set is open if it contains a particular point of the topological...

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...

particular point topology on the set {0,1}. The excluded point topology on any set with at least two elements is T0 but not T1. The only closed point...

different networks, yet their logical topologies may be identical. A network's physical topology is a particular concern of the physical layer of the OSI...

also called the connected two-point set − A 2-point set { 0 , 1 } {\displaystyle \{0,1\}} with the particular point topology { ∅ , { 1 } , { 0 , 1 } } ....

pointless topology, also called point-free topology (or pointfree topology) or topology without points and locale theory, is an approach to topology that avoids...

Finite particular point topology Countable particular point topology Uncountable particular point topology Sierpiński space, see also particular point topology...

In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks...

is relatively compact. In a non-Hausdorff space, such as the particular point topology on an infinite set, the closure of a compact subset is not necessarily...

In mathematics, the excluded point topology is a topology where exclusion of a particular point defines openness. Formally, let X be any non-empty set...

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

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

product topology or box topology, have the trivial topology. All sequences in X converge to every point of X. In particular, every sequence has a convergent...

operators acting on topological vector spaces, and the topology is often defined so as to capture a particular notion of convergence of sequences of functions...

mathematics, an order topology is a specific topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real...

of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology. The...

of an infinite discrete space. Any infinite set carrying the particular point topology is not paracompact; in fact it is not even metacompact. The Prüfer...

example of door space with more than one accumulation point is given by the particular point topology on a set X {\displaystyle X} with at least three points...

of both the finite particular point topology (with particular point 1) and the finite excluded point topology (with excluded point 0). Therefore, S {\displaystyle...

In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points in the closure of S not belonging to the...

that are commonly used in real or complex analysis; in particular, it is not Hausdorff. This topology was introduced primarily by Oscar Zariski and later...

with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the...

particularly in the field of topology, the K-topology, also called Smirnov's deleted sequence topology, is a topology on the set R of real numbers which...

Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function f...

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

the box topology has a somewhat more intuitive definition than the product topology, it satisfies fewer desirable properties. In particular, if all the...

In topology, filters can be used to study topological spaces and define basic topological notions such as convergence, continuity, compactness, and more...

topology of X ∪ P. For a set Z and a point p in Z, one obtains the particular point topology construction by considering in Z the discrete topology and...

theory is loosely regarded as a generalization of classical topology. Under meager point-set hypotheses, namely sobriety, this is completely accurate—it...

topology τ of a topological space (X, τ) is a family B {\displaystyle {\mathcal {B}}} of open subsets of X such that every open set of the topology is...