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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 an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms ...
This is a list of useful examples in general topology, a field of mathematics. Alexandrov topology; Cantor space; Co-kappa topology Cocountable topology; Cofinite topology; Compact-open topology; Compactification; Discrete topology; Double-pointed cofinite topology; Extended real number line; Finite topological space; Hawaiian earring; Hilbert cube
An open cover of a topological space X is a family of open sets U α such that their union is the whole space, U α = X. The order or ply of an open cover A {\displaystyle {\mathfrak {A}}} = { U α } is the smallest number m (if it exists) for which each point of the space belongs to at most m open sets in the cover: in other words U α 1 ∩ ...
Download as PDF; Printable version; In other projects ... Erdős space; List of examples in general topology; ... Fibration; Finite topological space; First ...
The following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics. This is not a list of properties that a topology or topological space might possess; for that, see List of general topology topics and Topological property.
The overlapping interval topology is a simple example of a topology that is T 0 but is not T 1. Every weakly Hausdorff space is T 1 but the converse is not true in general. The cofinite topology on an infinite set is a simple example of a topology that is T 1 but is not Hausdorff (T 2). This follows since no two nonempty open sets of the ...
A topological space is a set endowed with a structure, called a topology, which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces , and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology.
A closed subspace of a Baire space need not be Baire. See the Examples section. If a space contains a dense subspace that is Baire, it is also a Baire space. [17] A space that is locally Baire, in the sense that each point has a neighborhood that is a Baire space, is a Baire space. [18] [19] Every topological sum of Baire spaces is Baire. [20]