Thread: Intro to topology and open sets

Hello All,

Topology is just getting under way for me, and I have a question.

Consider a set {n}. Is it open, closed, or neither? Specifically, say {8}, for example.

I say this set is closed, because it has upper and lower bounds at {8}. Also, as I understand it, the compliment {(-inf,8) u (8, inf)} is open. Thoughts?

All input is greatly appreciated.

SY

Originally Posted by SYoungblood

Consider a set {n}. Is it open, closed, or neither? Specifically, say {8}, for example.
I say this set is closed, because it has upper and lower bounds at {8}. Also, as I understand it, the compliment {(-inf,8) u (8, inf)} is open. Thoughts?

With knowing the exact set of axioms in use, it is difficult to answer.
However, in the usual topology on the set of real numbers any finite set is closed.
The complement of a finite set is open so the set is closed.

Thank you kindly, my text didn't define discrete topology until the section after where I came across this little idea. I think I am good to go now. Relatively speaking.

Well, did your text define "topology"? It's impossible to define "open" or "closed" sets without having a topology and, for this question, it is necessary to say, explicitly, which topology, on this set, you are using.

Additionally, if you have $A\subseteq B$, then the subset is typically assumed to take the topology of the superset. However, if your full set (sometimes called a universe) is some set $A$, then regardless of the contents of $A$, if it has a topology, it is a clopen subset of itself.

Example, $A = [1,2] \cup (3,4] \cup [5,6) \cup (7,8) \cup \{9\}$ is neither an open nor a closed subset of the reals, but it is a clopen subset of itself in any topology (typically, the inherited topology from the reals).