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In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space is said to be metrizable if there is a metric such that the topology induced by is . Metrization theorems are theorems that give sufficient conditions for a topological space to be metrizable.
In mathematics, a topological space is called separable if it contains a countable, dense subset; that is, there exists a sequence of elements of the space such that every nonempty open subset of the space contains at least one element of the sequence.
This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundamental to algebraic topology, differential topology and geometric topology.
In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4: every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space. These conditions are examples of separation axioms and their further strengthenings define completely normal Hausdorff spaces, or T5 spaces, and perfectly normal Hausdorff spaces, or T6 spaces.
In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by Dieudonné (1944). Every compact space is paracompact. Every paracompact Hausdorff space is normal, and a Hausdorff space is paracompact if and only if it admits partitions of unity subordinate to any open cover. Sometimes paracompact spaces are defined so as to always be Hausdorff.
In mathematics, Tychonoff's theorem states that the product of any collection of compact topological spaces is compact with respect to the product topology. The theorem is named after Andrey Nikolayevich Tikhonov, who proved it first in 1930 for powers of the closed unit interval and in 1935 stated the full theorem along with the remark that its proof was the same as for the special case. The earliest known published proof is contained in a 1937 paper of Eduard Čech.
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.
In the mathematical field of topology, a Gδ set is a subset of a topological space that is a countable intersection of open sets. The notation originated in German with G for Gebiet meaning open set in this case and δ for Durchschnitt. The term inner limiting set is also used. Gδ sets, and their dual, Fσ sets, are the second level of the Borel hierarchy.
In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively studied by Polish topologists and logicians—Sierpiński, Kuratowski, Tarski and others. However, Polish spaces are mostly studied today because they are the primary setting for descriptive set theory, including the study of Borel equivalence relations. Polish spaces are also a convenient setting for more advanced measure theory, in particular in probability theory.
In topology, a second-countable space, also called a completely separable space, is a topological space whose topology has a countable base. More explicitly, a topological space is second-countable if there exists some countable collection of open subsets of such that any open subset of can be written as a union of elements of some subfamily of . A second-countable space is said to satisfy the second axiom of countability. Like other countability axioms, the property of being second-countable restricts the number of open sets that a space can have.
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 topology induced from that of X called the subspace topology.
In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space which is invariant under homeomorphisms. That is, a property of spaces is a topological property if whenever a space X possesses that property every space homeomorphic to X possesses that property. Informally, a topological property is a property of the space that can be expressed using open sets.
In topology, a branch of mathematics, a topological manifold is a topological space which locally resembles real n-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathematics. All manifolds are topological manifolds by definition. Other types of manifolds are formed by adding structure to a topological manifold. Every manifold has an "underlying" topological manifold, obtained by simply "forgetting" the added structure.
In mathematics, in the field of topology, a topological space is said to be collectionwise Hausdorff if given any closed discrete subset of , there is a pairwise disjoint family of open sets with each point of the discrete subset contained in exactly one of the open sets.
In mathematics, more specifically point-set topology, a Moore space is a developable regular Hausdorff space. That is, a topological space X is a Moore space if the following conditions hold:
In mathematics, a topological space is called collectionwise normal if for every discrete family Fi of closed subsets of there exists a pairwise disjoint family of open sets Ui, such that Fi ⊂ Ui. A family of subsets of is called discrete when every point of has a neighbourhood that intersects at most one of the sets from . An equivalent definition demands that the above Ui are themselves a discrete family, which is stronger than pairwise disjoint.
In mathematics, particularly topology, a Gδ space is a topological space in which closed sets are in a way ‘separated’ from their complements using only countably many open sets. A Gδ space may thus be regarded as a space satisfying a different kind of separation axiom. In fact normal Gδ spaces are referred to as perfectly normal spaces, and satisfy the strongest of separation axioms.
In general topology, a branch of mathematics, the Appert topology, named for Antoine Appert (1934), is a topology on the set X = Z+ = {1, 2, 3, …} of positive integers. In the Appert topology, the open sets are those that do not contain 1, and those that asymptotically contain almost every positive integer. The space X with the Appert topology is called the Appert space.
In mathematical set theory, the Cantor tree is either the full binary tree of height ω + 1, or a topological space related to this by joining its points with intervals, that was introduced by Robert Lee Moore in the late 1920s as an example of a non-metrizable Moore space.
References
- 1 2 Nyikos, Peter J. (2001), "A history of the normal Moore space problem", Handbook of the History of General Topology, Hist. Topol., 3, Dordrecht: Kluwer Academic Publishers, pp. 1179–1212, MR 1900271
- ↑ Jones, F. B. (1937), "Concerning normal and completely normal spaces", Bulletin of the American Mathematical Society , 43 (10): 671–677, doi: 10.1090/S0002-9904-1937-06622-5 , MR 1563615 .
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