Topology/Product Spaces

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Before we begin[edit]

We quickly review the set-theoretic concept of Cartesian product here. This definition might be slightly more generalized than what you're used to.

Cartesian Product[edit]

Definition[edit]

Let \Lambda be an indexed set, and let X_\lambda be a set for each \lambda \in \Lambda. The Cartesian product of each X_\lambda is


\prod_{\lambda \in \Lambda}X_\lambda = \{x:\Lambda\rightarrow\bigcup_{i \in I} X_i | x(\lambda) \in X_\lambda\}.

Example[edit]

Let \Lambda = \mathbb{N} and X_\lambda = \mathbb{R} for each n \in \mathbb{N}. Then


\prod_{\lambda \in \Lambda} X_\lambda = \mathbb{R}^\mathbb{N} = \{x: \mathbb{N} \rightarrow \mathbb{R} \mid x(n) \in \mathbb{R}\, \forall\, n \in \mathbb{N}\} = \{(x_1, x_2, \ldots) \mid x_n \in \mathbb{R}\, \forall\, n \in \mathbb{N}\}.

Product Topology[edit]

Using the Cartesian product, we can now define products of topological spaces.

Definition[edit]

Let X_\lambda be a topological space. The product topology of \prod_{\lambda \in \Lambda} X_\lambda is the topology with base elements of the form \prod_{\lambda \in \Lambda} U_\lambda, where U_\lambda = X_\lambda for all but a finite number of \lambda and each U_\lambda is open.

Examples[edit]

  • Let \Lambda = \{1,2\} and X_\lambda = \mathbb{R} with the usual topology. Then the basic open sets of \mathbb{R}^2 have the form (a,b) \times (c,d):

Opensetusual.jpg

  • Let \Lambda = \{1,2\} and X_\lambda = R_l (The Sorgenfrey topology). Then the basic open sets of \mathbb{R}^2 are of the form [a,b)\times [a,b):

Opensetsorg.jpg


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