About openness in topology From the Princeton book for the GRE Subject Test in Maths:





Questions:


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*Does 'this topology' in the first blue box refer to the topology in the red box?

*In the first blue box, what does it mean for a set to be open in a topology? I mean, should the book have omitted the word 'open' ? The book seemed to have proven that open intervals are members of the topology in the red box.

*In the second blue box, what does it mean for a set to be closed in a topology? I'm guessing it means the set's complement is in or is open in the topology (whatever that means) based on the second green box.

Notes:


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*I'm not using Princeton as a replacement for textbooks or Schaum's. I'm using Princeton as a guide.

*I have done and plan to do more practice exams.

*I know topology may be in only at most 2 questions on an exam.

$1.$  Yes, the blue an red boxes refer to the same topology on $\mathbb{R}$.
$2.$  The word open could very well have been omitted and nothing would be changed.  The elements of a topology (which is a collection of subsets of the given set) are called open sets, so saying that $U$ is open in a topology is somewhat redundant.
$3.$  To say a set is closed in a topology is somewhat an abuse of notation.  A topology is, strictly speaking, a collection of subsets. Nevertheless, when someone says a set $K$ is closed in a topology, what they really mean is that $K^c$ is a member of the topology.
