Trying to understand what it means for a sequence to be in a topological space.
What are the possible sequences in $(\{1, 2, 3, 4\}, \tau_{triv})$ and $(\{1, 2, 3, 4\}, \tau_{disc})$? Do the sequences depend on the underlying topology?
I am thinking the only possible sequence in the given trivial space above is $\{1, 2, 3, 4\}$, unless $\{\}$ is also a sequence whereas there are $2^4$ possible sequences in the given discrete space above, among which are $\{1\}, \{1, 4\}, \{2, 3, 4\}$. Does that make sense?
The reason I ask this question is the following two proofs below:
In $(X, \tau_{triv})$, let $l \in X$ and $(x_n)$ be any sequence. If $N$ is ‘any’ neighborhood of $l$ then $N$ must, in fact, be the whole of $X$. So $n \ge 1 \implies x_n \in N \implies x_n \to l.$
In $(X, \tau_{disc})$, suppose $(x_n)$ converges to $l$. Now $\{l\}$ is open and is a neighborhood of $l$, so $\exists n_0 \in N$ s.t. $n \ge n_0 \implies x_n \in \{l\} \implies x_n = l$. So $(x_n)$ is eventually constant (at $l$).
Why do they choose these particular neighborhoods in the proofs above? Likely because they are convenient for the conclusion. Then, why are these particular $N$ considered appropriately arbitrary? For example, do they choose $N$ to be $X$ in the first proof because there's possibly only one sequence ($\{1, 2, 3, 4\})$ for a limit to be in? What about $N = \{l\}$? Thanks.
Best Answer
If a topological space consists of a set $X$ and a topology $\tau$, a sequence in it is simply $(x_1,x_2,x_3,...),$ where $x_n\in X$ for all $n\in\mathbb N$. There are infinitely many sequences in $X=\{1,2,3,4\}$, even though $X$ has only four elements. As spaceisdarkgreen commented, the possible sequences have nothing to do with the topology. Whether or not a given sequence converges to a given point has to do with the topology. A sequence converges to $l$ if the elements of the sequence are eventually in any neighborhood of $l$. In the discrete topology, any subset of $X$ is open, so the elements of the sequence are eventually in any set containing $l$. In fact, $\{l\}$ itself is open, so the elements of the sequence are eventually in the singleton $\{l\}$; i.e., the sequence is eventually constant. For example, $(1,2,1,1,1,1,1,1,1,...)$ converges in the discrete and in the trivial topology, whereas $(1,2,3,1,2,3,1,2,3,...)$ converges in the trivial but not in the discrete topology.