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Let's go through out 2 examples regarding topology that i need help with.

The question we are looking at is: Classify the following sets as open, closed, clopen or neither in the standard $\mathbb{R}^{2}$ topology:

1.1 $(0, 1)$ x $[0, 1]$

1.2 $\bigcup_{n \ge 1} (0, 1 +\frac{1}{n})$ x $(-1, 1 - \frac{1}{n})$

I know that in order to prove that a set $U$ is open i can show that every point $x \in U$ has a open neighborhood $M$ such that $x \in M \subset U$. To prove that a set $U$ is closed I just need to prove that $U^{\complement}$ is open.

But unfortunatly i'm stuck when trying to figure out a formal proof regarding these two examples, if anyone could give me a hint based on basic topology knowledge on how should I start it I would appreciate a lot.

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  • $\begingroup$ Let me call your sets $A$ and $B$, respectively. $A$ is a product of an open set with a closed set, so your intuition should say you it is neither open nor closed. Fan you find some point for which every neighbourhood goes out of $A$? For $B$, I don't know if the union affects two or sets or just the first one, but it doesn't matter. It's either union of products of open sets or product of open sets so... $\endgroup$
    – Dog_69
    Apr 11, 2019 at 19:17
  • $\begingroup$ Thanks, it helped quite a lot. $\endgroup$
    – Yan R.
    Apr 11, 2019 at 19:20
  • $\begingroup$ Fantastic! I'm very glad then. By the way, I meant "Try to find some point for which..." $\endgroup$
    – Dog_69
    Apr 11, 2019 at 19:23
  • $\begingroup$ For $1.2$ you need to show that for each $n$, $U_n=(0,1+1/n)\times(-1,1-1/n)$ is open. This is trivial if the definition of the topology on $\mathbb R^2$ is the product topology, but a slight bit harder if it's based on the Euclidean metric (though still not very hard.) Once you have $U_n$ is open for each $n,$ prove that the union of any number of open sets is an open set. $\endgroup$ Apr 11, 2019 at 19:32
  • $\begingroup$ Thanks for the help, i think there is actually no need to prove that the union of any number of open sets is an open set since it quite follows from the definition of topology (An arbitrary union of open sets is open) so it's indeed easy to do. $\endgroup$
    – Yan R.
    Apr 11, 2019 at 19:44

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