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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).
The real line with its usual topology is a locally compact Hausdorff space; hence we can define a Borel measure on it. In this case, B ( R ) {\displaystyle {\mathfrak {B}}(\mathbb {R} )} is the smallest σ-algebra that contains the open intervals of R {\displaystyle \mathbb {R} } .
The little Bernshtein theorem: A function that is absolutely monotonic on a closed interval [,] can be extended to an analytic function on the interval defined by | | <. A function that is absolutely monotonic on [ 0 , ∞ ) {\displaystyle [0,\infty )} can be extended to a function that is not only analytic on the real line but is even the ...
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 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 ...
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.
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.