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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.
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 ...
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]
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.
Let (,) be a metric space, where is a given set. For any point and any non-empty subset , define the distance between the point and the subset: (,):= (,),.For any sequence of subsets {} = of , the Kuratowski limit inferior (or lower closed limit) of as ; is := {:,} = {: (,) =}; the Kuratowski limit superior (or upper closed limit) of as ; is := {:,} = {: (,) =}; If the Kuratowski limits ...
The finest topology on X is the discrete topology; this topology makes all subsets open. The coarsest topology on X is the trivial topology; this topology only admits the empty set and the whole space as open sets. In function spaces and spaces of measures there are often a number of possible topologies.
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance.More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms ...