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?

Best Answer

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...

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