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  2. Lower limit topology - Wikipedia

    en.wikipedia.org/wiki/Lower_limit_topology

    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).

  3. Borel measure - Wikipedia

    en.wikipedia.org/wiki/Borel_measure

    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} } .

  4. Sorgenfrey plane - Wikipedia

    en.wikipedia.org/wiki/Sorgenfrey_plane

    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 .

  5. Half-open interval topology - Wikipedia

    en.wikipedia.org/?title=Half-open_interval...

    Half-open interval topology. Add languages. Add links. Article; Talk; ... Download as PDF; Printable version ... move to sidebar hide. From Wikipedia, the free ...

  6. Lindelöf space - Wikipedia

    en.wikipedia.org/wiki/Lindelöf_space

    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.

  7. Open and closed maps - Wikipedia

    en.wikipedia.org/wiki/Open_and_closed_maps

    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 f : X → Y {\displaystyle f:X\to Y} is open if for any open set U {\displaystyle U} in X , {\displaystyle X,} the image f ( U ) {\displaystyle f(U)} is open in Y ...

  8. Radon measure - Wikipedia

    en.wikipedia.org/wiki/Radon_measure

    The measure M is outer regular, and locally finite, and inner regular for open sets. It coincides with m on compact and open sets, and m can be reconstructed from M as the unique inner regular measure that is the same as M on compact sets. The measure m is called moderated if M is σ-finite; in this case the measures m and M are the same.

  9. Derived set (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Derived_set_(mathematics)

    In mathematics, more specifically in point-set topology, the derived set of a subset of a topological space is the set of all limit points of . It is usually denoted by S ′ . {\displaystyle S'.} The concept was first introduced by Georg Cantor in 1872 and he developed set theory in large part to study derived sets on the real line .