Product Topology

**Drexel28** · February 9th 2010, 04:51 PM

Hello everyone. I have the following problem.

Let $X_j=[0,1]$ with the usual topology for every $j\in[0,1]$ and let $X=\prod_{j\in[0,1]}X_j$ . Prove that $X$ is not second countable.

The quote says to consider that if $X$ were second countable that every open base must have a countable subset which is also an open base.

But, I don't understand why this doesn't work.

Clearly we have that $G_j=\prod_{k\in[0,1]}E_k$ where $E_k=\begin{cases} (0,1) & \mbox{if} \quad k=j \\ X_k & \mbox{if} \quad k\ne j\end{cases}$ is open in $X$ (in fact it's a subbasic open set). So, we have that $O=\bigcup_{j\in[0,1]}G_j$ is an open set in $X$ . If $X$ were second countable we would have, by Lindelof's theorem, that $O$ must have a countable subcover. But, that isn't the case here. Right?

**Drexel28** · February 10th 2010, 03:22 PM

Originally Posted by Drexel28

Hello everyone. I have the following problem.

The quote says to consider that if $X$ were second countable that every open base must have a countable subset which is also an open base.

But, I don't understand why this doesn't work.

Clearly we have that $G_j=\prod_{k\in[0,1]}E_k$ where $E_k=\begin{cases} (0,1) & \mbox{if} \quad k=j \\ X_k & \mbox{if} \quad k\ne j\end{cases}$ is open in $X$ (in fact it's a subbasic open set). So, we have that $O=\bigcup_{j\in[0,1]}G_j$ is an open set in $X$ . If $X$ were second countable we would have, by Lindelof's theorem, that $O$ must have a countable subcover. But, that isn't the case here. Right?

Just as a follow-up. I posted this on the AOPS website and got an affirmative response.

Clearly $O$ is an open set engineered to have an uncountable open cover with no proper subcover. The fact that it cannot satisfy Lindelof's theorem, and thus cannot be second countable follows.

**mabruka** · February 10th 2010, 03:51 PM

Right, i agree it can not have a countable subcover.

Suppose there is, $O \subset \bigcup_{j=i_1,i_2,...}G_j$ for some numbers $\{i_1,i_2,...\} \subset [0,1]$

Now let $x \in X$ be such an element that $\pi_j (x) = 1$ for every $j \in \{i_1,i_2,...\}$ where $\pi_j$ is the projection on the jth coordenate.

Clearly $x \not \in\bigcup_{j=i_1,i_2,...}G_j$ but $x \in \bigcup_{j\in[0,1]}G_j = O$

What do you think?

**Drexel28** · February 10th 2010, 04:01 PM

Originally Posted by mabruka

Right, i agree it can not have a countable subcover.

Suppose there is, $O \subset \bigcup_{j=i_1,i_2,...}G_j$ for some numbers $\{i_1,i_2,...\} \subset [0,1]$

Now let $x \in X$ be such an element that $\pi_j (x) = 1$ for every $j \in \{i_1,i_2,...\}$ where $\pi_j$ is the projection on the jth coordenate.

Clearly $x \not \in\bigcup_{j=i_1,i_2,...}G_j$ but $x \in \bigcup_{j\in[0,1]}G_j = O$

What do you think?

Yes. I agree. Here is what I said. Let $\bold{x}_\ell$ be such that $\pi_\ell\left(\bold{x}_\ell\right)=\frac{1}{2}$ and $\pi_{k}\left(\bold{x}_j\right)=1,\text{ }k\ne \ell$ . Clearly we have that $\bold{x}_\ell$ is only in $G_\ell$ and so removing it from $\left\{G_j\right\}_{j\in[0,1]}$ will result in a cover that no longer covers $O$ . Since $\ell$ was arbitrary it follows that the removal of any set in $\left\{G_j\right\}_{j\in[0,1]}$ will result in a non-cover. Thus, $\left\{G_j\right\}_{j\in[0,1]}$ has no proper subcover and thus clearly no countable subcover.

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