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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]
For an ordered space (X, <) (i.e. a totally ordered set equipped with the order topology), the following are equivalent: (X, <) is compact. Every subset of X has a supremum (i.e. a least upper bound) in X. Every subset of X has an infimum (i.e. a greatest lower bound) in X. Every nonempty closed subset of X has a maximum and a minimum element.
The reverse implications do not hold. For example, the lower limit topology on the real line is first-countable, separable, and Lindelöf, but not second-countable. For metric spaces, however, the properties of being second-countable, separable, and Lindelöf are all equivalent. [1] Therefore, the lower limit topology on the real line is not ...
Ordinal-indexed sequences are more powerful than ordinary (ω-indexed) sequences to determine limits in topology: for example, ω 1 is a limit point of ω 1 +1 (because it is a limit ordinal), and, indeed, it is the limit of the ω 1-indexed sequence which maps any ordinal less than ω 1 to itself: however, it is not the limit of any ordinary ...