Скачать или смотреть The Indexed Family

The capital Greek letter Pi,
the product symbol, big Pi,
followed by X subscript I,
the product over i in I.

This set of functions
we call X,
that plain X we put
to the left.
(That plain X
we put to the left).

Recall the def. of X,
the set of all functions f,
with domain I, the index set,
to the union of the indexed sets,

such that for every index i
in the index set big I,
f of the index little i
is an element of X sub i.

For every index little i
we define projection pi
from the product set named X
to the factor X sub i.

For each element x in X,
the rule we now apply:
The value pi of x
equals x of little i.

To make a product space
we now apply topology,
the coarsest one that grants
each projection continuity.

The subbasis script 𝒮
gives us a neat supply,
the collection of inverse images
of open sets under pi.

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Written and Composed by: Aitlantis Civic (Atlantis Velforening)