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Rational sequence topology : ウィキペディア英語版
Rational sequence topology
In mathematics, more specifically general topology, the rational sequence topology is an example of a topology given to the set of real numbers, denoted R.
To give R a topology means to say which subsets of R are "open", and to do so in a way that the following axioms are met:
# The union of open sets is an open set.
# The finite intersection of open sets is an open set.
# R and the empty set ∅ are open sets.
== Construction ==

Let ''x'' be an irrational number (cf. rational number). Take a sequence of rational numbers with the property that converges, with respect to the Euclidean topology, towards ''x'' as ''k'' tends towards infinity. Informally, this means that each of the numbers in the sequence get closer and closer to ''x'' as we progress further and further along the sequence.
The rational sequence topology is given by defining both the whole set R and the empty set ∅ to be open, defining each rational number singleton to be open, and using as a basis for the irrational number ''x'', the sets 
: U_n(x) := \ \cup \ .

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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