Box topology

This article is about a basic definition in topology.
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Definition

Let $I$ be an indexing set, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ A_i \}_{i \in I}} be a family of topological spaces. Define:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P := \times_{i \in I} A_i}

to be the Cartesian product of the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_i} s. The box topology is a topology on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} in terms of the topologies on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_i} s defined in the following equivalent ways:

It is a topology with basis as the open boxes or open rectangles: sets of the form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \times_{i \in I} U_i} where each Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_i} is open in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_i}
Given a basis for each space Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_i} , it is a topology with basis as sets of the form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \times_{i \in I} U_i} , where each Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_i} is a basis element of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_i}

Equivalence of definitions

Further information: Equivalence of definitions of box topology, Open boxes satisfy the condition for a basis

Related notions

Product topology is a related, and more useful, topology on the Cartesian product of topological spaces. In fact, the default topology endowed on the Cartesian product of topological spaces is the product topology. The box topology and product topology are equal for products of only finitely many spaces. For infinite products, the product topology is a coarser topology, because it admits in its basis only those open rectangles where all but finitely many of the open sides are the whole space.

A list of properties of topological spaces closed under taking box products is available at Category:Properties of topological spaces closed under box products.

References

Textbook references

Topology (2nd edition) by James R. Munkres^{More info}, Page 114, Chapter 2, Section 19