Example of a point that is not the limit of any sequence in a connected topological space

Question

Question:

Let $X$ be a connected space with a topology not necessarily sequential. What is an example where a point in $X$ is not the limit of any not eventually constant sequence?

Motivation. Suppose $X$ is a topological vector space over $\mathbb{R}$ or $\mathbb{C}$. If we define a sequentially separated set $S$ of $X$ as such that, for every $x\in S$, $x$ lies outside the sequential closure of subspace $Y_x:=\text{span}(S\setminus \{x\})$. I'm trying to use the usual Zorn's lemma argument claiming there always exists a maximal such set. But it seems, if $X$ has a not-so-nice topology, there might be points which cannot be approximated by any not eventually constant sequence, and that can derail the reasoning.

I came to think about this issue when trying to understand uncountable Schauder basis. Thanks.

Answer 1

Let $X = \omega_1\times [0, 1) \cup \{\omega_1\}$ where $\omega_1\times [0, 1)$ is given lexicographic order and $\omega_1$ is a point which is greater than all points of $\omega_1\times [0, 1)$.

Give $X$ the order topology.

Then $X$ is connected and $\omega_1$ is not a limit of a not eventually constant sequence.

Imagine $X$ as a long closed interval, modification of the long ray to which we add a top element $\omega_1$, or as the ordinal $\omega_1+1$ to which we fill in the gaps between ordinals using segmemts.

Answer 2

When looking for pathological examples in topology (or just counterexamples, in general), it is usually best to start with very small, simple cases. Don't make your life too complicated—try to find the smallest space which has the properties you need. In this case, consider a two point set $\{a,b\}$ with the topology $\{\varnothing, \{a\}, \{a,b\}\}$.

• This space is connected, as there do not exist two nonempty disjoint open sets (let alone two nonempty disjoint open sets whose union is the entire space).

• Any sequence which has limit $a$ must eventually be constantly $a$.

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NB: As FShrike points out in the comments, this space is the Sierpinski space. I've never heard that term before, so I learned something new today. Huzzah.