discrete topology sentence in Hindi
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- 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.
- Since \ mathbb { T } is compact, the topology on the dual group is that of uniform convergence, which turns out to be the discrete topology.
- For instance, an example of a first-countable space which is not second-countable is counterexample # 3, the discrete topology on an uncountable set.
- Basic open sets in the discrete topology consist of individual letters; thus, the open cylinders of the product topology on S ^ \ mathbb { Z } are
- The discrete topology gives a functor \ mathbf { Set } \ to \ mathbf { Top }, where \ mathbf { Top } is the category of topological spaces.
- 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 ).
- In the case when " X " \ { " p " } has the discrete topology, the closed extension topology is the same as the particular point topology.
- 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 ).
- ;Prodiscrete topology : The prodiscrete topology on a product " A " " G " is the product topology when each factor " A " is given the discrete topology.
- 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the topology of above construction only relates to the indeterminate Y, since the topology that was put on \ ZX has been replaced by the discrete topology when defining the topology of the whole ring.
- 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.
- 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.
- One often considers " continuous group actions " : the group " G " is a topological group, " X " is a topological space, and the map is discrete topology.
- 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.
- 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.
- 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.
- 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.
- 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.
- 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 ).
- 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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