Induced topology

In topology and related areas of mathematics, an induced topology on a topological space is a topology that makes a given (inducing) function or collection of functions continuous from this topological space.

A coinduced topology or final topology makes the given (coinducing) collection of functions continuous to this topological space.

Definition

The case of just one function

Let [math]\displaystyle{ X_0, X_1 }[/math] be sets, [math]\displaystyle{ f:X_0\to X_1 }[/math].

If [math]\displaystyle{ \tau_0 }[/math] is a topology on [math]\displaystyle{ X_0 }[/math], then the topology coinduced on [math]\displaystyle{ X_1 }[/math] by [math]\displaystyle{ f }[/math] is [math]\displaystyle{ \{U_1\subseteq X_1 | f^{-1}(U_1)\in\tau_0\} }[/math].

If [math]\displaystyle{ \tau_1 }[/math] is a topology on [math]\displaystyle{ X_1 }[/math], then the topology induced on [math]\displaystyle{ X_0 }[/math] by [math]\displaystyle{ f }[/math] is [math]\displaystyle{ \{f^{-1}(U_1) | U_1\in\tau_1\} }[/math].

The easy way to remember the definitions above is to notice that finding an inverse image is used in both. This is because inverse image preserves union and intersection. Finding a direct image does not preserve intersection in general. Here is an example where this becomes a hurdle. Consider a set [math]\displaystyle{ X_0=\{-2, -1, 1, 2\} }[/math] with a topology [math]\displaystyle{ \{\{-2, -1\}, \{1, 2\}\} }[/math], a set [math]\displaystyle{ X_1=\{-1, 0, 1\} }[/math] and a function [math]\displaystyle{ f:X_0\to X_1 }[/math] such that [math]\displaystyle{ f(-2)=-1, f(-1)=0, f(1)=0, f(2)=1 }[/math]. A set of subsets [math]\displaystyle{ \tau_1=\{f(U_0)|U_0\in\tau_0\} }[/math] is not a topology, because [math]\displaystyle{ \{\{-1, 0\}, \{0, 1\}\} \subseteq \tau_1 }[/math] but [math]\displaystyle{ \{-1, 0\} \cap \{0, 1\} \notin \tau_1 }[/math].

There are equivalent definitions below.

The topology [math]\displaystyle{ \tau_1 }[/math] coinduced on [math]\displaystyle{ X_1 }[/math] by [math]\displaystyle{ f }[/math] is the finest topology such that [math]\displaystyle{ f }[/math] is continuous [math]\displaystyle{ (X_0, \tau_0) \to (X_1, \tau_1) }[/math]. This is a particular case of the final topology on [math]\displaystyle{ X_1 }[/math].

The topology [math]\displaystyle{ \tau_0 }[/math] induced on [math]\displaystyle{ X_0 }[/math] by [math]\displaystyle{ f }[/math] is the coarsest topology such that [math]\displaystyle{ f }[/math] is continuous [math]\displaystyle{ (X_0, \tau_0) \to (X_1, \tau_1) }[/math]. This is a particular case of the initial topology on [math]\displaystyle{ X_0 }[/math].

General case

Given a set X and an indexed family (Y_i)_i∈I of topological spaces with functions

[math]\displaystyle{ f_i: X \to Y_i, }[/math]

the topology [math]\displaystyle{ \tau }[/math] on [math]\displaystyle{ X }[/math] induced by these functions is the coarsest topology on X such that each

[math]\displaystyle{ f_i: (X,\tau) \to Y_i }[/math]

is continuous.

Explicitly, the induced topology is the collection of open sets generated by all sets of the form [math]\displaystyle{ f_i^{-1}(U) }[/math], where [math]\displaystyle{ U }[/math] is an open set in [math]\displaystyle{ Y_i }[/math] for some i ∈ I, under finite intersections and arbitrary unions. The sets [math]\displaystyle{ f_i^{-1}(U) }[/math] are often called cylinder sets. If I contains exactly one element, all the open sets of [math]\displaystyle{ (X,\tau) }[/math] are cylinder sets.

Examples

• The quotient topology is the topology coinduced by the quotient map.
• The product topology is the topology induced by the projections [math]\displaystyle{ \text{proj}_j : X \to X_j }[/math].
• If [math]\displaystyle{ f:X_0\to X }[/math] is an inclusion map, then [math]\displaystyle{ f }[/math] induces on [math]\displaystyle{ X_0 }[/math] the subspace topology.
• The weak topology is that induced by the dual on a topological vector space.

See also

• Natural topology
• The initial topology and final topology are used synonymously, though usually only in the case where the (co)inducing collection consists of more than one function.

Sources

• Hu, Sze-Tsen (1969). Elements of general topology. Holden-Day.