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1. Topological space: A topological space is a pair (X, T), where X is a set, and T is a collection of subsets of X that satisfy the following 3 axioms: Elements in T are called "open sets". [100%] 2023-06-28 [Topology]
2. Topological space: A totality of two elements: A set $X$, consisting of elements of an arbitrary nature, called points of the given space, and a topological structure, or topology, on this set $X$ (cf. Topological structure (topology)); it is immaterial whether this ... (Mathematics) [100%] 2023-10-25
3. Topological space: In mathematics, a topological space is an ordered pair ( X , T ) {\displaystyle (X,{\mathcal {T}})} where X {\displaystyle X} is a set and T {\displaystyle {\mathcal {T}}} is a certain collection of subsets of X {\displaystyle X} called the open ... [100%] 2023-06-30
4. Topological space: In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with ... (Mathematical space with a notion of closeness) [100%] 2024-01-12 [General topology] [Topological spaces]...
5. Category of topological spaces: In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous, and the ... [94%] 2023-12-21 [Categories in category theory] [General topology]...
6. Space of mappings, topological: A set $F$ of mappings from a set $X$ into a topological space $Y$ with some natural topology $\mathfrak{T}$ on $F$. For fixed $X$ and $Y$ one obtains different spaces of mappings, depending on which mappings $X \rightarrow Y ... (Mathematics) [87%] 2023-12-22
7. Axiomatic foundations of topological spaces: In the mathematical field of topology, a topological space is usually defined by declaring its open sets.({{{1}}}, {{{2}}}) However, this is not necessary, as there are many equivalent axiomatic foundations, each leading to exactly the same concept. For instance ... [84%] 2023-11-08 [General topology]
8. Topological vector space: In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is ... (Vector space with a notion of nearness) [81%] 2023-12-28 [Articles containing proofs] [Topology of function spaces]...
9. Noetherian topological space: In mathematics, a Noetherian topological space, named for Emmy Noether, is a topological space in which closed subsets satisfy the descending chain condition. Equivalently, we could say that the open subsets satisfy the ascending chain condition, since they are the ... [81%] 2024-01-12 [Algebraic geometry] [Properties of topological spaces]...
10. Topological vector space: over a topological field $ K $ A vector space $ E $ over $ K $ equipped with a topology (cf. Topological structure (topology)) that is compatible with the vector space structure, that is, the following axioms are satisfied: 1) the mapping $ ( x _ {1 ... (Mathematics) [81%] 2023-10-18
11. Pre-topological space: Let $X$ be a set and $\mathcal{P}X$ the set of subsets of $X$. A pre-topological space structure on $X$ is defined by a Čech closure operator, a mapping $C : \mathcal{P}X \rightarrow \mathcal{P}X$ such ... (Mathematics) [81%] 2023-11-14
12. Irreducible topological space: hyperconnected A topological space that cannot be represented as the union of two proper closed subspaces. Equivalently, an irreducible topological space can also be defined by postulating that any open subset of it is connected or that any non-empty ... (Mathematics) [81%] 2023-10-23 [General topology]
13. Topological vector space: In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is ... (Vector space with a notion of nearness) [81%] 2024-01-12 [Articles containing proofs] [Topology of function spaces]...
14. Finite topological space: In mathematics, a finite topological space is a topological space for which the underlying point set is finite. That is, it is a topological space which has only finitely many elements. [81%] 2023-11-23 [Topological spaces] [Combinatorics]...
15. Finite topological space: In mathematics, a finite topological space is a topological space for which the underlying point set is finite. That is, it is a topological space which has only finitely many elements. [81%] 2024-09-09 [Topological spaces] [Combinatorics]...
16. Extension of a topological space: A topological space $Y$ in which the given topological space $X$ is an everywhere-dense set. If $Y$ is a compact space, then it is called a compact extension, and if $Y$ is a Hausdorff space, it is called a ... (Mathematics) [78%] 2024-01-12 [General topology]
17. Tightness of a topological space: One of the cardinal characteristics of a topological space $X$. The local tightness $t(x,X)$ at a point $x \in X$ is the least cardinality $\mathfrak{t}\ge\aleph_0$ such that if $x$ is in the closure $\bar A ... (Mathematics) [78%] 2023-12-05
18. Dominant of a topological space: $X$ Any topological space for which $X$ serves as a retract (cf. Retract of a topological space). (Mathematics) [78%] 2024-01-11
19. Weight of a topological space: The smallest cardinal number which is the cardinality of an open base of a topological space. The weight, together with the cardinality, is the most important cardinal invariant of a topological space. (Mathematics) [78%] 2023-12-05
20. Characterizations of the category of topological spaces: In mathematics, a topological space is usually defined in terms of open sets. However, there are many equivalent characterizations of the category of topological spaces. [76%] 2023-10-31 [General topology]