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The Sorgenfrey line can thus be used to study right-sided limits: if : is a function, then the ordinary right-sided limit of at (when the codomain carries the standard topology) is the same as the usual limit of at when the domain is equipped with the lower limit topology and the codomain carries the standard topology.
Limit inferior is also called infimum limit, limit infimum, liminf, inferior limit, lower limit, or inner limit; limit superior is also known as supremum limit, limit supremum, limsup, superior limit, upper limit, or outer limit. An illustration of limit superior and limit inferior. The sequence x n is shown in blue.
Let (,) be a metric space, where is a given set. For any point and any non-empty subset , define the distance between the point and the subset: (,):= (,),.For any sequence of subsets {} = of , the Kuratowski limit inferior (or lower closed limit) of as ; is := {:,} = {: (,) =}; the Kuratowski limit superior (or upper closed limit) of as ; is := {:,} = {: (,) =}; If the Kuratowski limits ...
the lower limit topology or upper limit topology on the set R of real numbers (useful in the study of one-sided limits); any T 0, hence Hausdorff, topological vector space that is infinite-dimensional, such as an infinite-dimensional Hilbert space.
Upper topology; Scott topology. ... endowed with lower limit topology − It is Hausdorff, perfectly normal, first-countable, separable, paracompact, Lindelöf, ...
The real line can also be given the lower limit topology. Here, the basic open sets are the half open intervals [ a , b ) . {\displaystyle [a,b).} This topology on R {\displaystyle \mathbb {R} } is strictly finer than the Euclidean topology defined above; a sequence converges to a point in this topology if and only if it converges from above in ...
In mathematics, the upper topology on a partially ordered set X is the coarsest topology in which the closure of a singleton {} is the order section ] = {} for each . If ≤ {\displaystyle \leq } is a partial order, the upper topology is the least order consistent topology in which all open sets are up-sets .
The Γ-lower limit and the Γ-upper limit are defined as follows: ... the Kuratowski limes superior in the product topology of . In ...