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}

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