In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an ''N''-topological space is a set equipped with ''N'' arbitrary topologies. If ''τ''
1, ''τ''
2, ..., ''τ''
''N'' are ''N'' topologies defined on a nonempty set X, then the ''N''-topological space is denoted by (''X'',''τ''
1,''τ''
2,...,''τ''
''N'').
For ''N'' = 1, the structure is simply a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
.
For ''N'' = 2, the structure becomes a
bitopological space
In mathematics, a bitopological space is a set endowed with ''two'' topologies. Typically, if the set is X and the topologies are \sigma and \tau then the bitopological space is referred to as (X,\sigma,\tau). The notion was introduced by J. C. Kel ...
introduced by J. C. Kelly.
Example
Let ''X'' = be any finite set. Suppose ''A''
''r'' = . Then the collection ''τ''
1 = will be a topology on ''X''. If ''τ''
1, ''τ''
2, ..., ''τ''
''m'' be ''m'' such topologies (chain topologies) defined on ''X'', then the structure (''X'', ''τ''
1, ''τ''
2, ..., ''τ''
''m'') is an ''m''-topological space.
References
Mathematical terminology
Topology
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