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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 .
Counterexamples in Topology (1970, 2nd ed. 1978) is a book on mathematics by topologists Lynn Steen and J. Arthur Seebach, Jr. In the process of working on problems like the metrization problem , topologists (including Steen and Seebach) have defined a wide variety of topological properties .
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 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} } .
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.
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
The group V is obtained from T by adding the discontinuous map that fixes the points of the half-open interval [0,1/2) and exchanges [1/2,3/4) and [3/4,1) in the obvious way. On binary trees this corresponds to exchanging the two trees below the right-hand descendant of the root (if it exists).