This asymmetry disappears if the power series ring in Y is given the product topology where each copy of \ ZX is given its topology as a ring of formal power series rather than the discrete topology.
32.
These latter have the nice feature that the continuity of " r " is only with respect to the discrete topology on " V ", thus making the situation more algebraic in flavor.
33.
One often considers " continuous group actions " : the group " G " is a topological group, " X " is a topological space, and the map is discrete topology.
34.
Where the discrete topology is initial or free, the indiscrete topology is final or cofree : every function " from " a topological space " to " an indiscrete space is continuous, etc.
35.
This group ( with discrete topology ) can also be viewed as Pontryagin dual of \ operatorname { Gal } ( L / K ), assuming we regard \ mu _ n as a subgroup of circle group.
36.
Being Z a set and p a point in Z, one obtains the excluded point topology construction by considering in Z the discrete topology and applying the open extension topology construction to Z-{ p } plus p.
37.
Being Z a set and p a point in Z, one obtains the particular point topology construction by considering in Z the discrete topology and applying the closed extension topology construction to Z-{ p } plus p.
38.
The group algebra, consisting of " finite " sums, corresponds to functions on the group that vanish for cofinitely many points; topologically ( using the discrete topology ), these correspond to functions with compact support.
39.
In some ways, the opposite of the discrete topology is the trivial topology ( also called the " indiscrete topology " ), which has the fewest possible open sets ( just the empty set and the space itself ).
40.
The Cantor set is homeomorphic to the product of countably many copies of the discrete space { 0, 1 } and the space of irrational numbers is homeomorphic to the product of countably many copies of the natural numbers, where again each copy carries the discrete topology.
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