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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.
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.
Sorgenfrey line, which is endowed with lower limit topology − It is Hausdorff, perfectly normal, first-countable, separable, paracompact, Lindelöf, Baire, and a Moore space but not metrizable, second-countable, σ-compact, nor locally compact.
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 ...
The reverse implications do not hold, for example, standard Euclidean space (R n) is σ-compact but not compact, [5] and the lower limit topology on the real line is Lindelöf but not σ-compact. [6] In fact, the countable complement topology on any uncountable set is Lindelöf but neither σ-compact nor locally compact. [7]
The real line can also be given the lower limit topology. Here, the basic open sets are the half open intervals [a, b). This topology on 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 the Euclidean topology. This example shows that a set ...
Shqip; Sicilianu; Simple English ... In mathematics, a limit is the value that a function ... In topology, limits are used to define limit points of a subset of a ...
An example of T 0 space that is limit point compact and not countably compact is =, the set of all real numbers, with the right order topology, i.e., the topology generated by all intervals (,). [4] The space is limit point compact because given any point , every < is a limit point of {}.