# Product Space – Exercise Set 1

This post presents several exercises concerning product spaces. All the concepts involved in the exercises have been discussed in the blog.

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Exercise 1

Exercise 1a
Prove or disprove:
If $X$ and $Y$ are both hereditarily separable, then $X \times Y$ is hereditarily separable.

Exercise 1b
Show that if each $X_\alpha$ is separable, then the product space $\prod_{\alpha < \omega} \ X_\alpha$ is separable.

Exercise 1c
Prove or disprove:
If each $X_\alpha$ is separable, then the product space $\prod_{\alpha < \omega_1} \ X_\alpha$ is not separable.

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Exercise 2

Exercise 2a
Show that if the space $X$ is normal, then every closed subspace of $X$ is a normal space.

Exercise 2b
Prove or disprove:
If the space $X$ is normal, then every dense open subspace of $X$ is a normal space.

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Exercise 3

Consider the product space $\prod_{\alpha \in W} \ X_\alpha$.

Exercise 3a
Suppose that $X_\alpha$ is compact for all but one $\alpha \in W$ such that the non-compact factor is a Lindelof space. Show that the product space $\prod_{\alpha \in W} \ X_\alpha$ is a normal space.

Exercise 3b
Prove or disprove:
Suppose that $X_\alpha$ is compact for all but one $\alpha \in W$ such that the non-compact factor is a normal space. Then the product space $\prod_{\alpha \in W} \ X_\alpha$ is a normal space.

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Exercise 4

Exercise 4a
Let $X$ be a compact space.
Show that if $X^n$ is hereditarily Lindelof for all positive integer $n$, then $X$ is metrizable.

Exercise 4b
Prove or disprove:
If $X^n$ is hereditarily Lindelof for all positive integer $n$, then $X$ is metrizable.

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Exercise 5

Let $Y$ the product of uncountably many copies of the real line $\mathbb{R}$. If a specific example is desired, try $Y=\mathbb{R}^{\omega_1}$ ($\omega_1$ many copies of $\mathbb{R}$) or $Y=\mathbb{R}^{\mathbb{R}}$ (continuum many copies of $\mathbb{R}$). It is also OK to use a larger number of copies of the real line.

Note that the space $Y$ is not normal (see here).

Exercise 5a
Since the product space $Y$ is not normal, it is not Lindelof. As an exercise, find an open cover of $Y$ that proves that $Y$ is not Lindelof, i.e. an open cover $\mathcal{U}$ of $Y$ such that no countable subcollection of $\mathcal{U}$ can cover $Y$.

Exercise 5b
Show that for every open cover $\mathcal{U}$ of the space $Y$, there is a countable $\mathcal{V} \subset \mathcal{U}$ of $Y$ such that $\overline{\mathcal{V}}=Y$, i.e. $\cup \mathcal{V}$ is dense in $Y$. Note that with this property, the space $Y$ is said to be weakly Lindelof.

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Exercise 6

This exercise is about the product $Y=\mathbb{R}^{\mathbb{R}}$ (continuum many copies of $\mathbb{R}$). Show the following.

1. Show that $Y$ is separable by exhibiting a countable dense set.
2. Show that $Y$ is not hereditarily separable by exhibiting a non-separable subspace.
3. Show that the space $Y$ has a closed and discrete subspace of cardinality continuum.
4. Show that $Y$ is not first countable.
5. Show that $Y$ is not a Frechet space.
6. Show that $Y$ is not a countably tight space.

See here for the definition of Frechet space.

See here for the definition of countably tight space.

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Exercise 7

Consider the product space $Y=\mathbb{\omega}^{\omega_1}$. It is not normal (see here).

Exercise 7a
Construct a dense normal subspace of $Y$.

Exercise 7b
Construct a dense Lindelof subspace of $Y$.

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$\copyright \ 2016 \text{ by Dan Ma}$