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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).
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 .
Excluded point topology − A topological space where the open sets are defined in terms of the exclusion of a particular point. Fort space; Half-disk topology; Hilbert cube − [, /] [, /] [, /] with the product topology. Infinite broom; Integer broom topology; K-topology
This function from the unit circle to the half-open interval [0,2π) is bijective, open, and closed, but not continuous. It shows that the image of a compact space under an open or closed map need not be compact. Also note that if we consider this as a function from the unit circle to the real numbers, then it is neither open nor closed.
Half-open interval topology. Add languages. Add links. ... Upload file; Special pages ... Cite this page; Get shortened URL; Download QR code; Print/export Download ...
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.
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.
In the branch of mathematics known as topology, the topologist's sine curve or Warsaw sine curve is a topological space with several interesting properties that make it an important textbook example. It can be defined as the graph of the function sin(1/ x ) on the half-open interval (0, 1], together with the origin, under the topology induced ...