Difference between revisions of "Product topology"
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<math>p_\alpha:\prod_{\beta\in I}X_\beta\rightarrow X_\alpha</math> which take the [[Tuple|tuple]] <math>(x_\alpha)_{\alpha\in I}\rightarrow x_{\beta}</math> | <math>p_\alpha:\prod_{\beta\in I}X_\beta\rightarrow X_\alpha</math> which take the [[Tuple|tuple]] <math>(x_\alpha)_{\alpha\in I}\rightarrow x_{\beta}</math> | ||
| − | This leads to the main property of the product topology, which can best be expressed as a diagram. | + | This leads to the main property of the product topology, which can best be expressed as a diagram. As shown below: |
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| + | 10 x 5 = 50! this is a product | ||
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{{Todo}} | {{Todo}} | ||
{{Definition|Topology}} | {{Definition|Topology}} | ||
Revision as of 07:36, 23 August 2015
Given a set [ilmath]X_{\alpha\in I} [/ilmath] of indexed topological spaces, we define the product topology, denoted [math]\prod_{\alpha\in I}X_\alpha[/math] (yes the Cartesian product) is the coarsest topology such that all the projection maps are continuous.
The projection maps are:
[math]p_\alpha:\prod_{\beta\in I}X_\beta\rightarrow X_\alpha[/math] which take the tuple [math](x_\alpha)_{\alpha\in I}\rightarrow x_{\beta}[/math]
This leads to the main property of the product topology, which can best be expressed as a diagram. As shown below:
10 x 5 = 50! this is a product
TODO:
