Induced topology

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In topology and related areas of mathematics, an induced topology on a topological space is a topology which is "optimal" for some function from/to this topological space.

Definition[edit]

Let X_0, X_1 be sets, f:X_0\to X_1.

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

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

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 X_0=\{-2, -1, 1, 2\} with a topology \{\{-2, -1\}, \{1, 2\}\}, a set X_1=\{-1, 0, 1\} and a function f:X_0\to X_1 such that f(-2)=-1, f(-1)=0, f(1)=0, f(2)=1. A set of subsets \tau_1=\{f(U_0)|U_0\in\tau_0\} is not a topology, because \{\{-1, 0\}, \{0, 1\}\} \subseteq \tau_1 but \{-1, 0\} \cap \{0, 1\} \notin \tau_1.

There are equivalent definitions below.

A topology \tau_1 induced on X_1 by f is the finest topology such that f is continuous (X_0, \tau_0) \to (X_1, \tau_1). This is a particular case of the final topology on X_1.

A topology \tau_0 induced on X_0 by f is the coarsest topology such that f is continuous (X_0, \tau_0) \to (X_1, \tau_1). This is a particular case of the initial topology on X_0.

Examples[edit]

References[edit]

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

See also[edit]