Basis Element of the Product topology vs. Element of the Basis

elementary-set-theorygeneral-topology

Hi i am reading Topology by Munkres and have some doubts under theorem 20.3 regarding basis element of product topology. It is written that:

Let $B=(a_1,b_1)\times…\times(a_n,b_n)$ be a basis element for the product topology and let $\bar x =(x_1,…,x_n)$ be an element of $B$.

My doubts are as follows:

  1. I am unable to understand the above quoted statement. What does the notation $(a_1,b_1)\times(a_2,b_2)$ stand for?

  2. What is the difference between basis element $B$ and an element of $B$? I mean a basis element for the product topology on the cartesian product was defined by $U\times V$ previously in the book where U and V are open sets in X and Y respectively. Then why is it now $(a_1,b_1)\times(a_2,b_2)$ ? And how is it different from the tuple x =(x_1,…,x_n)?

For reference i am attaching the screenshot where i have highlighted the quoted text.Screenshot 1

Best Answer

  1. The $\times$ refers to the Cartesian product. Each $(a_i, b_i)$ is an (open) interval in $\mathbb{R}$. To visualise this, this is a box in the plane without its borders. So, for example, $$ (a_1, b_1) \times (a_2, b_2) = \{ (x_1, x_2) \in \mathbb{R}^2 : a_1 < x_1 < b_1, a_2 < x_2 < b_2 \} . $$ This pattern is then generalised to hold for $\mathbb{R}^n$.

  2. A basis for a topology means that every set in the topology can be generated as a union of elements of the basis (these elements are themselves sets). The product topology for $\mathbb{R}^n$ is defined as the one generated by these so-called cylindrical sets - an arbitrary such set is $B = (a_1, b_1) \times \dots \times (a_n, b_n)$. Since $B$ is itself a set (it is the Cartesian product of the $n$ open intervals), that means it contains points (if it is not empty), such as $x = (x_1, \dots, x_n)$.

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