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By definition, every σ-algebra, every filter (and so in particular, every neighborhood filter), and every topology is a covering π-system and so also a base for a topology. In fact, if Γ is a filter on X then { ∅ } ∪ Γ is a topology on X and Γ is a basis for it.
Example: The Hippocratic school held that four humors: blood, black bile, yellow bile and phlegm consists the basis for the four types of temperaments. Example: Kretschmer's classification system was based on three main body types: asthenic/leptosomic (thin, small, weak), athletic (muscular, large–boned), and pyknic (stocky, fat).
In topology, a subbase (or subbasis, prebase, prebasis) for a topological space with topology is a subcollection of that generates , in the sense that is the smallest topology containing as open sets. A slightly different definition is used by some authors, and there are other useful equivalent formulations of the definition; these are ...
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 ...
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 ...
Likewise, the neighborhood-based axioms (in the context of Hausdorff spaces) can be retraced to Felix Hausdorff's original definition of a topological space in Grundzüge der Mengenlehre. [citation needed] Many different textbooks use many different inter-dependences of concepts to develop point-set topology.
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 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 ...