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

____________________________________________________________________

**Exercise 1**

*Exercise 1a*

Prove or disprove:

If and are both hereditarily separable, then is hereditarily separable.

*Exercise 1b*

Show that if each is separable, then the product space is separable.

*Exercise 1c*

Prove or disprove:

If each is separable, then the product space is not separable.

____________________________________________________________________

**Exercise 2**

*Exercise 2a*

Show that if the space is normal, then every closed subspace of is a normal space.

*Exercise 2b*

Prove or disprove:

If the space is normal, then every dense open subspace of is a normal space.

____________________________________________________________________

**Exercise 3**

Consider the product space .

*Exercise 3a*

Suppose that is compact for all but one such that the non-compact factor is a Lindelof space. Show that the product space is a normal space.

*Exercise 3b*

Prove or disprove:

Suppose that is compact for all but one such that the non-compact factor is a normal space. Then the product space is a normal space.

____________________________________________________________________

**Exercise 4**

*Exercise 4a*

Let be a compact space.

Show that if is hereditarily Lindelof for all positive integer , then is metrizable.

*Exercise 4b*

Prove or disprove:

If is hereditarily Lindelof for all positive integer , then is metrizable.

____________________________________________________________________

**Exercise 5**

Let the product of uncountably many copies of the real line . If a specific example is desired, try ( many copies of ) or (continuum many copies of ). It is also OK to use a larger number of copies of the real line.

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

*Exercise 5a*

Since the product space is not normal, it is not Lindelof. As an exercise, find an open cover of that proves that is not Lindelof, i.e. an open cover of such that no countable subcollection of can cover .

*Exercise 5b*

Show that for every open cover of the space , there is a countable of such that , i.e. is dense in . Note that with this property, the space is said to be weakly Lindelof.

____________________________________________________________________

**Exercise 6**

This exercise is about the product (continuum many copies of ). Show the following.

- Show that is separable by exhibiting a countable dense set.
- Show that is not hereditarily separable by exhibiting a non-separable subspace.
- Show that the space has a closed and discrete subspace of cardinality continuum.
- Show that is not first countable.
- Show that is not a Frechet space.
- Show that is not a countably tight space.

See here for the definition of Frechet space.

See here for the definition of countably tight space.

____________________________________________________________________

**Exercise 7**

Consider the product space . It is not normal (see here).

*Exercise 7a*

Construct a dense normal subspace of .

*Exercise 7b*

Construct a dense Lindelof subspace of .

____________________________________________________________________