discrete topology sentence in Hindi
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- The natural topology on the set of fixed-length strings or variable-length strings is the discrete topology, but the natural topology on the set of infinite strings is the limit topology, viewing the set of infinite strings as the inverse limit of the sets of finite strings.
- More precisely, for any cardinal ?, there is a complete Boolean algebra of cardinality 2 ? greater than ? that is generated as a complete Boolean algebra by a countable subset; for example the Boolean algebra of regular open sets in the product space ? ?, where ? has the discrete topology.
- In the theory of locally compact topological groups the Pr�fer " p "-group ( endowed with the discrete topology ) is the Pontryagin dual of the compact group of p-adic integers, and the group of " p "-adic integers is the Pontryagin dual of the Pr�fer " p "-group.
- Since [ " T " ] is a closed set for the product topology on ? ? ?? ? where ? and ? are equipped with the discrete topology ( and all closed sets in ? ? ?? ? come in this way from some tree on ? ?? ), ?-Suslin subsets of ? ? are projections of closed subsets in ? ? ?? ?.
- For this reason this topology is called the "'open extension topology "'of X plus P, with which one extends to X * " P the open sets of X . Note that the subspace topology of X as a subset of X * " P is the original topology of X, while the subspace topology of P as a subset of X * " P is the discrete topology.
- For this reason this topology is called the "'closed extension topology "'of X plus P, with which one extends to X * " P the closed sets of X . Note that the subspace topology of X as a subset of X * " P is the original topology of X, while the subspace topology of P as a subset of X * " P is the discrete topology.
- For these reasons this topology is called the "'extension topology "'of X plus P, with which one extends to X * " P the open and the closed sets of X . Note that the subspace topology of X as a subset of X * " P is the original topology of X, while the subspace topology of P as a subset of X * " P is the discrete topology.
- For example, the product of the unit circle ( with its usual topology ) and the real line with the discrete topology is a locally compact group with the product topology and Haar measure on this group is not inner regular for the closed subset { 1 } x [ 0, 1 ] . ( Compact subsets of this vertical segment are finite sets and points have measure 0, so the measure of any compact subset of this vertical segment is 0.
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