nLab weak topology

Induced topologies

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

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Induced topologies

Definitions

See also at Top the section Universal constructions.

Weak/coarse/initial topology

Suppose

  1. SS is a set,

  2. {(X i,T i)} iI\{ (X_i, T_i) \}_{i \in I} is a family of topological spaces

  3. {f i:SX i} iI\big\{ f_i \colon S \to X_i \big\}_{i \in I} an indexed set of functions from SS to the family {X i} iI\{ X_i \}_{i \in I}.

Let Γ\Gamma denote the set of all topologies τ\tau on SS such that f if_i is a continuous map for every iIi \in I. Then the intersection τΓτ\bigcap_{\tau \in \Gamma} \tau is again a topology and also belongs to Γ\Gamma. Clearly, it is the coarsest/weakest topology τ 0\tau_0 on XX such that each function f i:SX if_i\colon S \to X_i is a continuous map.

We call τ 0\tau_0 the weak/coarse/initial topology induced on SS by the family of mappings {f i} iI\{ f_i \}_{i \in I}. Note that all terms ‘weak topology’, ‘initial topology’, and ‘induced topology’ are used. The subspace topology is a special case, where II is a singleton and the unique function f if_i is an injection.

Strong/fine/final topology

Dually, suppose

  1. SS is a set,

  2. {(X i,T i)} iI\{ (X_i, T_i) \}_{i \in I} a family of topological spaces

  3. {f i:X iS} iI\big\{ f_i \colon X_i \to S \big\}_{i \in I} a family of functions to SS from the family {X i} iI\{ X_i \}_{i \in I}.

Let Γ\Gamma denote the set of all topologies τ\tau on SS such that f if_i is a continuous map for every iIi \in I. Then the union τΓτ\bigcup_{\tau \in \Gamma} \tau is again a topology and also belongs to Γ\Gamma. Clearly, it is the finest/strongest topology τ 0\tau_0 on SS such that each function f i:X iSf_i\colon X_i \to S is a continuous map.

We call τ 0\tau_0 the strong/fine/final topology induced on SS by the family of mappings {f i} iI\{ f_i \}_{i \in I}. Note that all terms ‘strong topology’, ‘final topology’, and ‘induced topology’ are used. The quotient topology is a special case, where II is a singleton and the unique function f if_i 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 VV is a topological vector space over the ground field KK, then we may consider the continuous linear functionals on VV, that is the continuous linear maps from VV to KK. Taking VV to be the set XX in the general definition above, taking each T iT_i to be KK, and taking the continuous linear functionals on VV to comprise the family of functions, then we get the weak topology on VV.

The weak-star topology on the dual space V *V^* of continuous linear functionals on VV is precisely the weak topology induced by the dual (evaluation) functionals on V *V^*

{V *ev vK, by ff(v)} vV. \left\{V^* \overset{\operatorname{ev}_v}{\to} K, \text{ by } f \mapsto f(v)\right\}_{v \in V}.

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

Last revised on April 30, 2023 at 07:59:30. See the history of this page for a list of all contributions to it.