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The lower limit topology is finer (has more open sets) than the standard topology on the real numbers (which is generated by the open intervals). The reason is that every open interval can be written as a (countably infinite) union of half-open intervals.
The definition of a point of closure of a set is closely related to the definition of a limit point of a set.The difference between the two definitions is subtle but important – namely, in the definition of a limit point of a set , every neighbourhood of must contain a point of other than itself, i.e., each neighbourhood of obviously has but it also must have a point of that is not equal to ...
An open (resp. closed) map f : X → Y remains open (resp. closed) if the topology on Y becomes finer or the topology on X coarser. One can also compare topologies using neighborhood bases . Let τ 1 and τ 2 be two topologies on a set X and let B i ( x ) be a local base for the topology τ i at x ∈ X for i = 1,2.
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.
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.
An open cover of a space ... Every closed subset of a paracompact space is paracompact. ... The square of the real line R in the lower limit topology is a classical ...
In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counterintuitive, as the common meanings of open and closed are antonyms, but their mathematical definitions are not mutually exclusive .
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]