# Sequentially closed but no closed subset?

I am given a product topology of uncountable $$X_a$$ (R as index) each homeomorphic to [0, 1]. The product toplogical space is compact and normal but not a metric space. So the question is, is there a subset of the product topological space that is not closed but sequentially closed? I do not think so, but just want to make sure.

• Hint: consider sequences of ones and zeros. May 7 '20 at 15:05

There are such sets. A good hint why such sets exist is that despite $$[0,1]$$ being first countable, uncountable products usually do not preserve first countability, and non-first countable spaces have a good chance of being non-sequential - but that is not sufficient.

A concrete example is the set $$A:= \{(x_i)_{i \in \mathbb{R}} \vert x_i = 1 ~ \text{for countably many} ~ i, x_i = 0 ~ \text{otherwise}\}$$. Then $$A$$ is sequentially closed, but not closed. To show this, we show that an arbitrary convergent sequence in $$A$$ has its limit point in $$A$$, but there are convergent nets, whose limit point does not lie in $$A$$.

Then let $$(x^n)_{n \in \mathbb{N}} \subseteq A$$ be an arbitrary convergent sequence with limit $$y \in [0,1]^{\mathbb{R}}$$, then the set of $$i$$s s.t. $$y_i = 1$$ is countable. Recall that $$y_i = 1$$ is by definition equivalent to $$x_i^n = 1$$ for large enough $$n$$. Hence

$$\{i \in \mathbb{R} : y_i = 1\} \subseteq \{i \in \mathbb{R} : x_i^n = 1, n \geq m, m \in \mathbb{N}\} = \cup_{m \in \mathbb{N}} \cap_{n \geq m}^{\infty} \{i : x_i^n = 1\}$$

where the $$\{i : x_i^n = 1\}$$ is countable for each $$n \in \mathbb{N}$$ by assumption.

However, consider the net $$(x^j)_{j \in \mathbb{R}}$$ indexed by $$\mathbb{R}$$ defined by $$x^j_i = 1$$ when $$i \leq j$$ and $$0$$ otherwise, then this net converges to constant $$1$$ which is not in $$A$$. Hence this net has a limit which is not in $$A$$ and thus $$A$$ is not closed.

• umm so A is not closed since (1,1,1,1,1,1,1,1,1,1....) is an accumulation point? May 8 '20 at 0:25
• also I'm having hard time understanding why y has to have countable 1 coordinates May 8 '20 at 1:40