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In mathematics, the lower limit topology or right half-open interval topology is a topology defined on , the set of real numbers; it is different from the standard topology on (generated by the open intervals) and has a number of interesting properties.
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.
In mathematical analysis, limit superior and limit inferior are important tools for studying sequences of real numbers.Since the supremum and infimum of an unbounded set of real numbers may not exist (the reals are not a complete lattice), it is convenient to consider sequences in the affinely extended real number system: we add the positive and negative infinities to the real line to give the ...
Consider the set K of all functions f : → [0, 1] from the real number line to the closed unit interval, and define a topology on K so that a sequence {} in K converges towards f ∈ K if and only if {()} converges towards f(x) for all real numbers x.
Extended real number line; ... endowed with lower limit topology − It is Hausdorff, perfectly normal, first-countable, separable, paracompact, Lindelöf, ...
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 ...
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 ...
Continuum (topology) Extended real number line; Long line (topology) Sierpinski space; Cantor set, Cantor space, Cantor cube; Space-filling curve; Topologist's sine curve; Uniform norm; Weak topology; Strong topology; Hilbert cube; Lower limit topology; Sorgenfrey plane; Real tree; Compact-open topology; Zariski topology; Kuratowski closure ...