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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.
5 Examples. 6 Uniform spaces. 7 Metric spaces. 8 Topology and order theory. 9 Descriptive set theory. ... Lower limit topology; Sorgenfrey plane; Real tree; Compact ...
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 following is a list of named topologies or topological spaces, many of which are counterexamples in topology and related branches of mathematics. This is not a list of properties that a topology or topological space might possess; for that, see List of general topology topics and Topological property.
In carrying the lower limit topology, no uncountable set is compact. In the cocountable topology on an uncountable set, no infinite set is compact. Like the previous example, the space as a whole is not locally compact but is still Lindelöf. The closed unit interval [0, 1] is compact.
Equivariant topology; Erdős space; List of examples in general topology; ... Long line (topology) Loop space; Lower limit topology; M. Menger sponge; Metric space;
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 ...