About openness in topology

Question

From the Princeton book for the GRE Subject Test in Maths:

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Questions:

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

2. 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.

3. 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.

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Notes:

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

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

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

Answer 1

The answer to your first question is "yes."

For your second and third questions: Phrases like "$A$ is [open/closed/Borel/etc.] in the topology $\tau$" use the word "in" in a perhaps odd way - think "in the context of," or "according to," or "in the sense of." For example:

• "$A$ is open in $\tau$" means exactly "$A\in\tau$."

• "$A$ is closed in $\tau$" means exactly "$A^c\in\tau$" (where "$^c$" denotes the complement).

• "$A$ is not open in $\tau$" means "$A\not\in\tau$".

• And so on.

Answer 2

1. Yes, 'this topology' is the topology referred to in the red box. This is the standard topology on $\mathbb{R}$.
2. When someone says that a set is "open in a topology," that means that it is a member of the topology. Every set in your topology $T$ is called an open set, as mentioned in the green box.
3. A set is closed if its complement is open (i.e. if its complement is in the topology).

A little bit of unsolicited advice: It seems like you haven't studied point-set topology too much before. I know that you mentioned not using Princeton as a replace for textbooks, but it kind of seems like you're learning from it as if it's a textbook. I think you'd be much better off reading some of the basics in Munkres to get an idea of what's going on.

Answer 3

$1.$ 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.

Answer 4

1. Yes, 'this topology' does refer to the topology described in the red box. It is commonly known as the standard topology on $\mathbb R$. Note however that there are open sets in this topology that are not open intervals, e.g. $(0,1) \cup (1,2)$. You can prove that every open set in the standard topology of $\mathbb R$ is a union of open intervals (it's an easy exercise).
2. The word 'open' is historically motivated. Take a look at the standard topology of $\mathbb R$. Every set in this topology (i.e. every open set with respect to this topology) is a union of open intervals. The concept of a topology has been abstracted from this (and similar) topologies and there are examples of topologies whose members don't look 'open' in an intuitive manner. In fact, there are examples that don't permit an easy geometrical interpretation. Take for example the toplogy generated by $n$-types.
3. Let $X$ be a nonempty set and let $\tau$ be a topology on $X$. I.e. $\tau \subseteq P(X)$ is such that
   • $\emptyset, X \in \tau$,
   • $O_0, O_1 \in \tau \rightarrow O_o \cap O_1 \in \tau$ and
   • $S \subseteq \tau \rightarrow \bigcup S \in \tau$.

(Note that this is the exact same definition as given in the text - modulo a slightly different notation.) Then a set $C \subseteq X$ is closed if and only if $X \setminus C \in \tau$. Again - this definition is motivated by the geometrical intuition provided by the standard topology on $\mathbb R$. However, similiar to open sets, in general this geometrical intuition breaks down and being closed is nothing but a shorthand for that fact that the complement is in the topology at hand.