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This question already has an answer here:

[Similar to this, but I think the question I'm asking isn't covered]

This is part of a problem in section 9.2 of S. Morris's "Topology Without Tears".

I'm confused about the following construction:

Let $(X_{i}, \tau_{i})$ be $T1$ $\forall i \in \mathbb{N}$.

Then, the product space $\prod_{i \in \mathbb{N}}(X_i, \tau_i)$ should also be $T1$.

In a $T1$ space, all singletons [i.e., points of the form $(a_1,...,a_n,...)$] are closed.

Hence, the complement of a singleton is open.

(**)Thus, the complement of a singleton in this space is of the form $\prod_{i \in \mathbb{N}}O_i$ where $O_i\in \tau_i$ and $O_i\not= X_i$.

Question: Shouldn't this open set be impossible in the Product Topology?

[Edit: Answer below, thanks to William Elliot. The problem is with the statement (**).]

Morris defines the Product Topology in such a way that you can't form open sets of the type described in (**). Specifically, in a basis set there can only be a finite number of factors subsets that are not equal to the entire factor set (i.e., only a finite number of $O_i\not= X_i$).

I understand that we might not be using the product topology in this problem, since it's not the only common topology for infinite products (cf. here).

But if that's so, how am I supposed to know when I should be using the Product Topology or some other topology? Am I supposed to assume the former as a default unless I run into trouble and then switch?

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marked as duplicate by Paul Frost, Henno Brandsma general-topology Mar 10 at 15:57

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The complement is
$\cup_{n \in N} (X_1×...×X_{n-1}×(X_n - \{a_n\})×X_{n+1}×...)$.

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  • $\begingroup$ Ohhhh...I see what I was doing. Thanks! $\endgroup$ – Cassius12 Mar 11 at 2:03

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