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I am very new to Topology. I do not have a college math degree, only highschool math. I have a real hard time understanding how a set can be both open and closed. And set being neither open nor closed.

Is it correct to think of a set being both open and closed as below?

Let's consider a topology over two sets (A and B) that are disjoint. For example, A = (1,2) and B = (4,5). Both the sets are open as they don't contain their limit points. However, by definition, a set is closed if that set's compliment is open. Therefore, A is closed because A's compliment (which is B) is open and B is closed because its compliment is open.

I am yet to come up with an example for sets that are neither open nor closed. It would be really great if someone can give a simple example for that.

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  • $\begingroup$ The rationals are neither open nor closed. Their interior is empty, their closure is all of $\mathbb{R}$. $\endgroup$ Dec 17, 2020 at 1:45
  • $\begingroup$ The empty set and the entire set are both open and closed. And in this context, complement does not have an i $\endgroup$ Dec 17, 2020 at 1:45
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    $\begingroup$ Compliment: “Your post is well crafted and you provide appropriate context and evidence of work. Well done!”. Complement: the stuff that is not part of that being considered. $\endgroup$ Dec 17, 2020 at 1:46
  • $\begingroup$ @ArturoMagidin thanks. I was misreading the question. $\endgroup$
    – Rob Arthan
    Dec 17, 2020 at 1:49
  • $\begingroup$ When you talk about sets being open, closed, or neither, it's important to be clear about which topology you mean. To take your example, $A$ is clopen in the subspace topology induced by the usual topology on $\Bbb R$ on the subspace (of $\Bbb R$) $A \cup B$, but $A$ is merely open (not also closed) considered as a subspace of $\Bbb R$. $\endgroup$ Dec 17, 2020 at 2:13

2 Answers 2

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In the real numbers $\mathbb{R}$, any half-open interval of the form $[a,b)$ or $(a,b]$ where $a<b$ is neither open nor closed.

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The definition of an open set in the real line is a set that is equal to the set of its inner points, that is: a set $A$ is open if for every $a\in A$ there exist a small number $\varepsilon$ such that $(a-\varepsilon,a+\varepsilon)$ is contained in $A$, a set is closed if its complement is open. Take $A=[0,1)$ in $\mathbb R$, this set is not open since $(0-\varepsilon,0+\varepsilon)$ will never be contained in $A$ but $0\in A$. And it's complement is not open since $1\in A^c$ but $(1-\varepsilon,1+\varepsilon)$ cant be contained in $A^c$, so $[0,1)$ is neither open or closed.

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