For example, any group can be considered as a topological group by giving it the discrete topology, implying that theorems about topological groups apply to all groups.
22.
Since \ mathbb { T } is compact, the topology on the dual group is that of uniform convergence, which turns out to be the discrete topology.
23.
For instance, an example of a first-countable space which is not second-countable is counterexample # 3, the discrete topology on an uncountable set.
24.
Basic open sets in the discrete topology consist of individual letters; thus, the open cylinders of the product topology on S ^ \ mathbb { Z } are
25.
The discrete topology gives a functor \ mathbf { Set } \ to \ mathbf { Top }, where \ mathbf { Top } is the category of topological spaces.
26.
Also directly from the definition, an aspherical space is the classifying space of its fundamental group ( considered to be a topological group when endowed with the discrete topology ).
27.
In the case when " X " \ { " p " } has the discrete topology, the closed extension topology is the same as the particular point topology.
28.
In the two extremes, every set can be open ( called the discrete topology ), or no set can be open but the space itself and the empty set ( the indiscrete topology ).
29.
;Prodiscrete topology : The prodiscrete topology on a product " A " " G " is the product topology when each factor " A " is given the discrete topology.
30.
Then the topology of above construction only relates to the indeterminate Y, since the topology that was put on \ ZX has been replaced by the discrete topology when defining the topology of the whole ring.
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