# Sequences convergence in different topological spaces.

general-topologysolution-verification

In different topological spaces, sequential convergence can be really bizarre. The following are 6 examples:

1. $$X$$ is an uncountable set with the cofinite topology (in which the closed sets are $$X$$ and finite subsets of $$X$$).
2. $$X$$ is any uncountable set with co-countable topology, in which the closed sets are $$X$$ and all countable subsets of $$X$$.
3. $$X$$ is the real line with the topology in which the open sets are the sets of the form $$(a,\infty), a \in \mathbf{R}$$
4. $$X$$ is the sorgenfrey Line $$\mathbf{E}$$
5. $$X$$ is a discrete space.
6. $$X$$ is any trivial space.

My question is, how to find/exhaust all convergent sequences? For 1,2, there is a complete answer here. So we don't care about 1,2 here. We only discuss 3-6 here.

For 3: I think for any $$x \in X$$, the convergent sequence to $$x$$ is any sequence that finally lies in $$(x,\infty)$$

For 4: We know the Sorgenfrey line $$\mathbf{E}$$ is finer than usual topology $$\tau$$ on $$\mathbf{R}$$. So sequences converging in $$\mathbf{E}$$ must also converge in $$\mathbf{R}$$ . I only find from a lecture that: "Decreasing sequences which are bounded below converge in the Sorgenfrey line" However, is that all?

For 5: The only convergent sequence is eventually a constant sequence.

For 6: Any sequence is convergent to any point.

Am I doing right in examples 3-6?

You're right about $$3,5,6$$, but for $$3$$ you'll need a small argument. And you have to be careful about how you state it: you want to characterise a sequence that converges to some point, and the characterisation would (IMO) not mention a limit at all. Just properties of the sequence.
For the Sorgenfrey line it's clear that if $$(x_n)$$ is Sorgenfrey convergent it is also Euclidean-convergent (to the same limit), and almost all terms lie on the right of that limit...