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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} } .
Download as PDF; Printable version; ... Let the real line have its standard topology. Then every open subset of the real line is a countable union of open intervals.
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.
Half-open interval topology. Add languages. Add links. Article; ... Download QR code; Print/export Download as PDF; Printable version; In other projects Appearance.
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.
Concretely, a connected compact 1-manifold with boundary is an interval = [,] or a circle (compactness rules out the open interval (,) and the half-open interval [,), neither of which yields non-trivial embeddings since the open end means that they can be shrunk to a point), so a possibly disconnected compact 1-manifold is a collection of n ...