Search results
Results from the WOW.Com Content Network
The lower limit topology is finer (has more open sets) than the standard topology on the real numbers (which is generated by the open intervals). The reason is that every open interval can be written as a (countably infinite) union of half-open intervals. For any real and , the interval [,) is clopen in (i.e., both open and closed).
In topology, the Sorgenfrey plane is a frequently-cited counterexample to many otherwise plausible-sounding conjectures. It consists of the product of two copies of the Sorgenfrey line , which is the real line R {\displaystyle \mathbb {R} } under the half-open interval topology .
In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. [1] [2] [3] That is, a function : is open if for any open set in , the image is open in . Likewise, a closed map is a function that maps closed sets to closed sets.
If is endowed with its usual Euclidean topology then the derived set of the half-open interval [,) is the closed interval [,]. Consider R {\displaystyle \mathbb {R} } with the topology (open sets) consisting of the empty set and any subset of R {\displaystyle \mathbb {R} } that contains 1.
In summary, a set of the real numbers is an interval, if and only if it is an open interval, a closed interval, or a half-open interval. [4] [5] A degenerate interval is any set consisting of a single real number (i.e., an interval of the form [a, a]). [6] Some authors include the empty set in this definition.
Half-open interval topology. Add languages. Add links. ... Upload file; Special pages ... Cite this page; Get shortened URL; Download QR code; Print/export Download ...
Let X be the interval [0, 1) equipped with the topology generated by the collection of half open intervals {[a, b) : 0 ≤ a < b ≤ 1}. This topology is sometimes called Sorgenfrey line. On this topological space, standard Lebesgue measure is not Radon since it is not inner regular, since compact sets are at most countable.
The usual example of this is the Sorgenfrey plane, which is the product of the real line under the half-open interval topology with itself. Open sets in the Sorgenfrey plane are unions of half-open rectangles that include the south and west edges and omit the north and east edges, including the northwest, northeast, and southeast corners.