Axiomatic foundations of topological spaces - Wikipedia
In the mathematical field of topology, a topological space is usually defined by declaring its open sets. [1] However, this is not necessary, as there are many equivalent axiomatic foundations, each leading to exactly the same concept.
Separation axiom - Wikipedia
The separation axioms are about the use of topological means to distinguish disjoint sets and distinct points. It's not enough for elements of a topological space to be distinct (that is, unequal); we may want them to be topologically distinguishable.
Topological space - Wikipedia
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 formalizing the concept of closeness.
The topological space (X,T ) is called the prime spectrum of the ring A and is denoted Spect(A). The closure of a one point set {x} in Spec(A) consists of all prime ideals y ∈ X =Spec(A) containing x.
Separation Axioms -- from Wolfram MathWorld
Oct 10, 2024 · Separation Axioms. A list of five properties of a topological space expressing how rich the "population" of open sets is. More precisely, each of them tells us how tightly a closed subset can be wrapped in an open set.
Hausdorff Axioms -- from Wolfram MathWorld
Oct 10, 2024 · The axioms formulated by Hausdorff (1919) for his concept of a topological space. These axioms describe the properties satisfied by subsets of elements x in a neighborhood set E of x. 1. There corresponds to each point x at least one neighborhood U (x), and each neighborhood U (x) contains the point x. 2.
A topological space (X; T ) is said to be T1 if for any pair of distinct points x; y 2 X, there exist open sets U and V such that U contains x but not y, and V contains y but not x. Recall that an equivalent de nition of a T1 space is one in which all singletons are closed.
topological space is a pair (X; T ) where X consists of a set of objects called points and T is a collection of subsets of X called open sets such that the following are satis ed: ; 2 T and X 2 T . If A 2 T and B 2 T then A \ B 2 T . If U T then [fuju 2 Ug 2 T . The collection T is called the topology of X. The statement \(X; T ) is.
T_1-Space -- from Wolfram MathWorld
Oct 10, 2024 · A T_1-space is a topological space fulfilling the T1-separation axiom: For any two points x,y in X there exists two open sets U and V such that x in U and y not in U, and y in V and x not in V.
Definition 2.1 A space X is a T0 space iff it satisfies the T0 axiom, i.e. for each x, y ∈ X such that x 6= y there is an open set U ⊂ X so that U contains one of x and y but not the other. Obviously the property T0 is a topological property. An arbitrary product of T0 spaces is T0.