This question is from "Introduction to Topology: Pure and Applied," by Colin Adams and Robert Franzosa.
Here's how the authors define a topology:
Let X be a set. A topology Ƭ on X is a collection of subsets of X, each called an open set, such that
(i) ∅ and X are open sets;
(ii) The intersection of finitely many open sets is an open set;
(iii) The union of any collection of open sets is an open set.
Here's how they define a discrete topology:
Let X be a nonempty set and let Ƭ be the collection of all subsets of X. Clearly this is a topology, since unions and intersections of subsets of X are themselves subsets of X and therefore are in the collection Ƭ. We call this the discrete topology on X. This is the largest topology that we can define on X.
Here's where I am with this problem:
First, is this what it's asking? Prove that Ƭ is a discrete topology on X if and only if every x in X is a set {x} in Ƭ?
If that is indeed the question, I'm still struggling. (I'm very bad at proofs).
If we look at the definition of a discrete topology, it seems self evident. A discrete topology must contain all subsets of X, which would include every x in X. I just have no idea how to write a mathematical proof of that.
Appreciate any help.
Best Answer
Proof writing is as obvious as you think it is, just need to write it down formally
($\implies$) Assume $\mathcal{T}$ is the discrete topology. Then every subset $S$ of $X$ is open, ie. $S \in \mathcal{T}$. This includes all sets containing just a singleton, ie. all sets of the form $\{ x \}$ for $ x\in X$
($\impliedby$) Assume $\{ x \} \in \mathcal{T}$ for every $ x\in X$. Then notice that for any subset $A \subseteq X$, $A$ has (possibly infinite) elements who are members of $X$, namely $$ A = \{ x_i | x_i \in X \mbox{ and } i\in I \mbox{ where } I \mbox{ is an indexing set} \} $$ But this is exactly a union of singleton sets, $$ A = \bigcup_{i\in I} \{ x_i \} $$ which is open as the union of open sets.
Finally, if $A=\emptyset$ then it open as the intersection of 2 different open sets, ie. $\emptyset = \{ x_1 \} \cap \{ x_2 \}$ where $x_1 \neq x_2$.
If $A=X$, it is simply the union of all singleton sets of elements in $X$ hence open