The topology on the adelic algebraic group G ( A ) is taken to be the subspace topology in " A " " N ", the Cartesian product of " N " copies of the adele ring.
42.
The topology correctly assigned to the idele group is that induced by inclusion in " A " 2; composing with a projection, it follows that the ideles carry a finer topology than the subspace topology from " A ".
43.
A subset " F " of a topological space " X " is called irreducible or reducible, if " F " considered as a topological space via the subspace topology has the corresponding property in the above sense.
44.
Two important special cases of the initial topology are the product topology, where the family of functions is the set of projections from the product to each factor, and the subspace topology, where the family consists of just one function, the inclusion map.
45.
Subsets of the real line \ mathbb { R } ( with the induced subspace topology ) holding selection principles properties, most notably Menger and Hurewicz spaces, can be characterized by their continuous images in the Baire space \ mathbb { N } ^ \ mathbb { N }.
46.
A " paracompactifying " family of supports that satisfies further that any " Y " in ? is, with the subspace topology, a paracompact space; and has some " Z " in ? which is a Hausdorff the family of all compact subsets satisfies the further conditions, making it paracompactifying.
47.
Suppose by way of contradiction that there is some strict total order < on Z such that the order topology generated by < is equal to the subspace topology on Z ( note that we are not assuming that < is the induced order on Z, but rather an arbitrarily given total order on Z that generates the subspace topology ).
48.
Suppose by way of contradiction that there is some strict total order < on Z such that the order topology generated by < is equal to the subspace topology on Z ( note that we are not assuming that < is the induced order on Z, but rather an arbitrarily given total order on Z that generates the subspace topology ).
49.
Separable metrizable spaces can also be characterized as those spaces which are homeomorphic to a subspace of the Hilbert cube \ lbrack 0, 1 \ rbrack ^ \ mathbb { N }, i . e . the countably infinite product of the unit interval ( with its natural subspace topology from the reals ) with itself, endowed with the product topology.
50.
You seem to be confused about the definition of the subspace topology : if " U " is any open subset of \ mathbb R ^ 2, then U-\ mathbb Q ^ 2 is an open set in \ mathbb R ^ 2-\ mathbb Q ^ 2 . J . 17 : 33, 9 June 2009 ( UTC)
