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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 cofinite topology on an infinite set is locally compact in senses (1), (2), and (3), and compact as well, but it is not Hausdorff or regular so it is not locally compact in senses (4) or (5). The indiscrete topology on a set with at least two elements is locally compact in senses (1), (2), (3), and (4), and compact as well, but it is not ...
Every compact space is σ-compact, and every σ-compact space is Lindelöf (i.e. every open cover has a countable subcover). [4] 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]
Throughout the history of topology, connectedness and compactness have been two of the most widely studied topological properties. Indeed, the study of these properties even among subsets of Euclidean space, and the recognition of their independence from the particular form of the Euclidean metric, played a large role in clarifying the notion of a topological property and thus a topological space.
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.
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.
In mathematics, topology is a branch of geometry concerned with the study of topological spaces. The term topology is also used for a set of open sets used to define topological spaces. See the topology glossary for common terms and their definition. Properties of general topological spaces (as opposed to manifolds) are discussed in general ...
The study of Top and of properties of topological spaces using the techniques of category theory is known as categorical topology. N.B. Some authors use the name Top for the categories with topological manifolds , with compactly generated spaces as objects and continuous maps as morphisms or with the category of compactly generated weak ...