weak topology in nLab
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For the strong topology in functional analysis, see at strong operator topology.
Context
Topology
topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
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fiber space, space attachment
Extra stuff, structure, properties
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Kolmogorov space, Hausdorff space, regular space, normal space
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sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
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closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
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open subspaces of compact Hausdorff spaces are locally compact
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compact spaces equivalently have converging subnet of every net
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continuous metric space valued function on compact metric space is uniformly continuous
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paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
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injective proper maps to locally compact spaces are equivalently the closed embeddings
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locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
Induced topologies
Definitions
See also at Top the section Universal constructions.
Weak/coarse/initial topology
Suppose
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is a set,
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is a family of topological spaces
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an indexed set of functions from to the family .
Let denote the set of all topologies on such that is a continuous map for every . Then the intersection is again a topology and also belongs to . Clearly, it is the coarsest/weakest topology on such that each function is a continuous map.
We call the weak/coarse/initial topology induced on by the family of mappings . Note that all terms ‘weak topology’, ‘initial topology’, and ‘induced topology’ are used. The subspace topology is a special case, where is a singleton and the unique function is an injection.
Strong/fine/final topology
Dually, suppose
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is a set,
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a family of topological spaces
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a family of functions to from the family .
Let denote the set of all topologies on such that is a continuous map for every . Then the union is again a topology and also belongs to . Clearly, it is the finest/strongest topology on such that each function is a continuous map.
We call the strong/fine/final topology induced on by the family of mappings . Note that all terms ‘strong topology’, ‘final topology’, and ‘induced topology’ are used. The quotient topology is a special case, where is a singleton and the unique function is a surjection.
Generalisations
We can perform the first construction in any topological concrete category, where it is a special case of an initial structure for a source or cosink.
We can also perform the second construction in any topological concrete category, where it is a special case of an final structure for a sink.
In functional analysis
In functional analysis, the term ‘weak topology’ is used in a special way. If is a topological vector space over the ground field , then we may consider the continuous linear functionals on , that is the continuous linear maps from to . Taking to be the set in the general definition above, taking each to be , and taking the continuous linear functionals on to comprise the family of functions, then we get the weak topology on .
The weak-star topology on the dual space of continuous linear functionals on is precisely the weak topology induced by the dual (evaluation) functionals on
For the strong topology in functional analysis, see the strong operator topology.
References
The original version of this article was posted by Vishal Lama at induced topology.
See also
- Wikipedia, Initial topology, Final topology,
Last revised on April 30, 2023 at 07:59:30. See the history of this page for a list of all contributions to it.