Gδ set

In the mathematical field of topology, a G_δ set is a subset of a topological space that is a countable intersection of open sets. The notation originated in Germany with G for Gebiet (German: area, or neighbourhood) meaning open set in this case and δ for Durchschnitt (German: intersection). The term inner limiting set is also used. G_δ sets, and their dual F_σ sets, are the second level of the Borel hierarchy.

Definition

In a topological space a G_δ set is a countable intersection of open sets. The G_δ sets are exactly the level sets of the Borel hierarchy.

Examples

Group action

In mathematics, a symmetry group is an abstraction used to describe the symmetries of an object. A group action formalizes the relationship between the group and the symmetries of the object. It relates each element of the group to a particular transformation of the object.

In this case, the group is also called a permutation group (especially if the set is finite or not a vector space) or transformation group (especially if the set is a vector space and the group acts like linear transformations of the set). A permutation representation of a group G is a representation of G as a group of permutations of the set (usually if the set is finite), and may be described as a group representation of G by permutation matrices. It is the same as a group action of G on an ordered basis of a vector space.

A group action is an extension to the notion of a symmetry group in which every element of the group "acts" like a bijective transformation (or "symmetry") of some set, without being identified with that transformation. This allows for a more comprehensive description of the symmetries of an object, such as a polyhedron, by allowing the same group to act on several different sets of features, such as the set of vertices, the set of edges and the set of faces of the polyhedron.

Sydney Trains T set

The Tangara (an Aboriginal Australian word meaning to go) is a class of electric multiple unit operated by Sydney Trains in Sydney, Australia. The Tangaras were delivered between 1988 and 1995, and are third-generation trains.

Design

A Tangara is a double-deck four-car set, with the two outer cars being driving control trailers fitted with one pantograph each and the middle two cars being non-control motor cars. They are equipped with air conditioning and chopper control.

Two subclasses of Tangara were built, the suburban sets targeted as T sets, and outer-suburban sets targeted as G sets. The T sets replaced the first generation of Sydney's electric rolling stock.

Unlike most other Sydney Trains trains the seats on the T sets are fixed, meaning that half the seats face backwards.

The G sets differed from the T sets in having yellow front panels, round green door buttons, high-backed reversible seats, toilets, fresh water dispensers and luggage racks.

Set G7 was fitted with an AC drive system for evaluation purposes with the existing DC stock and compatibility with signalling and communication systems on the network. G7 was scrapped in 2005 at Maintrain, Auburn after the Waterfall train disaster, as all four cars were beyond repair.