Club set

In mathematics, particularly in mathematical logic and set theory, a club set is a subset of a limit ordinal which is closed under the order topology, and is unbounded (see below) relative to the limit ordinal. The name club is a contraction of "closed and unbounded".

Formal definition

Formally, if is a limit ordinal, then a set is closed in if and only if for every , if , then . Thus, if the limit of some sequence from is less than , then the limit is also in .

If is a limit ordinal and then is unbounded in if for any , there is some such that .

If a set is both closed and unbounded, then it is a club set. Closed proper classes are also of interest (every proper class of ordinals is unbounded in the class of all ordinals).

For example, the set of all countable limit ordinals is a club set with respect to the first uncountable ordinal; but it is not a club set with respect to any higher limit ordinal, since it is neither closed nor unbounded. The set of all limit ordinals is closed unbounded in ( regular). In fact a club set is nothing else but the range of a normal function (i.e. increasing and continuous).

Above and Beyond

Above and Beyond may refer to:

See also

Set

Set or The Set may refer to:

Mathematics and computer programming

Physics

Chemistry

Psychology

Technology

Club Mix

Club Mix is a two-disc album remixed by British musician/DJ Sonique and released in 2001.

Track listing

Disc one

Disc two

Club Dance

Club Dance was a TV show that aired on The Nashville Network from April 1, 1991 to February 5, 1999 for a total of 1,848 episodes, with re-runs until June 28, 1999

The show was filmed in Knoxville, Tennessee set at a fictional "White Horse Cafe". The show was hosted by Shelley Mangrum, a former Miss Tennessee, and co-hosted by Phil Campbell. The cast consisted of enthusiastic unpaid local dancers and visiting professional dancers and dance groups from all over the United States, creating a unique atmosphere of mixing of styles and traditions. Within the limits of the script, the dancers were free to behave and respond to the music as if in a real dance venue.

The show included a wide variety types of country dances lead and follow partner dances, choreographed dances, and line dances, as well as some others, most notably West Coast Swing, which due its slotted style allowed for dancing between tables and in other confined spaces.

The show was accompanied with the monthly newsletter Club Dance: Behind the Scenes. It was produced by Knoxville-based RIVR Media.