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For example, the Euclidean topology on the plane admits as a base the set of all open rectangles with horizontal and vertical sides, and a nonempty intersection of two such basic open sets is also a basic open set. But another base for the same topology is the collection of all open disks; and here the full (B2) condition is necessary.
A three-dimensional model of a figure-eight knot.The figure-eight knot is a prime knot and has an Alexander–Briggs notation of 4 1.. Topology (from the Greek words τόπος, 'place, location', and λόγος, 'study') is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling ...
The term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed. This is a list of topology topics. See also: Topology glossary; List of topologies; List of general topology topics; List of geometric topology topics
A typical example is a surface such as the one depicted at the right. Moreover, is a point in X called the base-point. (As is explained below, its role is rather auxiliary.) The idea of the definition of the homotopy group is to measure how many (broadly speaking) curves on X can be deformed into each other. The precise definition depends on ...
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
Hence the point space of a locally compact connected Laguerre plane is homeomorphic to the cylinder or it is a -dimensional manifold, cf. [64] A large class of -dimensional examples, called ovoidal Laguerre planes, is given by the plane sections of a cylinder in real 3-space whose base is an oval in .
For every topological space , the identity map: is a covering. Likewise for any discrete space the projection : taking (,) is a covering. Coverings of this type are called trivial coverings; if has finitely many (say ) elements, the covering is called the trivial -sheeted covering of .
A more complicated example is the -adic topology on a ring and its modules. Let I {\displaystyle I} be an ideal of a ring R . {\displaystyle R.} The sets of the form x + I n {\displaystyle x+I^{n}} for all x ∈ R {\displaystyle x\in R} and all positive integers n , {\displaystyle n,} form a base for a topology on R {\displaystyle R} that makes ...