nLab
weak topology

Induced topologies

Definitions

Suppose SS is a set, {(X i,T i)} iI\{ (X_i, T_i) \}_{i \in I} a family of topological spaces and {f i} iI\{ f_i \}_{i \in I} a family of functions from SS to the family {X i} iI\{ X_i \}_{i \in I}. That is, for each index iIi \in I, f i:SX if_i\colon S \to X_i. Let Γ\Gamma be 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.

Dually, suppose SS is a set, {(X i,T i)} iI\{ (X_i, T_i) \}_{i \in I} a family of topological spaces and {f i} iI\{ f_i \}_{i \in I} a family of functions to SS from the family {X i} iI\{ X_i \}_{i \in I}. That is, for each index iIi \in I, f i:X iSf_i\colon X_i \to S. Let Γ\Gamma be 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 finest/strongest topology τ 0\tau_0 on XX 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 sink.

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? is another special case of a weak topology.

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.

Revised on October 12, 2013 21:30:01 by Toby Bartels (98.19.41.253)