Examples of Lindelof Spaces that are not Hereditarily Lindelof

We observe from the following statement two examples of Lindelof spaces that are not hereditarily Lindelof.

  • Any product space contains a discrete subspace having the same cardinality as the number of factor spaces.

Using the above observation, by choosing the factor spaces judiciously, the product of uncountably many spaces is a handy way of obtaining Lindelof spaces (in some cases \sigma-compact spaces) that are not hereditarily Lindelof. For definition and basic information about product spaces, see this previous post.

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All spaces under consideration are at least Hausdorff. For each \alpha \in A, let X_\alpha be a space with at least two points. For each \alpha \in A, fix two points p_\alpha, q_\alpha \in X_\alpha. Then the product space \displaystyle \prod \limits_{\alpha \in A} X_\alpha contains a discrete subspace Y that has the same cardinality as the cardinality of the index set A.

For each \alpha \in A, define y_\alpha \in \prod \limits_{\alpha \in A} X_\alpha by the following:

\displaystyle (1) \ \ \ \ \ \ y_\alpha(\gamma)=\left\{\begin{matrix}p_\alpha&\ \gamma=\alpha\\{q_\alpha}&\ \gamma \ne \alpha \end{matrix}\right.

Let Y=\left\{ y_\alpha: \alpha \in A\right\}. It follows that \lvert Y \lvert = \lvert A \lvert and that Y is a discrete space.

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Whenever the index set A is uncountable, the product space \displaystyle \prod \limits_{\alpha \in A} X_\alpha contains an uncountable discrete subspace. Thus even if the product space \displaystyle \prod \limits_{\alpha \in A} X_\alpha is Lindelof, one of its subspace Y cannot be Lindelof. Taking the product of uncountably many factor spaces is a handy way to obtain Lindelof space that is not hereditarily Lindelof. Some examples are shown below.

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Examples

Let the index set A be uncountable. To make the product space Lindelof, we can make every one of its factor X_\alpha compact. Thus the product space \displaystyle \prod \limits_{\alpha \in A} X_\alpha is compact and not hereditarily Lindelof.

Thus the product space [0,1]^{\omega_1}, the product of \omega_1 many copies of the unit interval, is compact and not hereditarily Lindelof. Another example is \left\{ 0,1 \right\}^{\omega_1}, the product of \omega_1 many copies of \left\{ 0,1 \right\}

Another way to make the product space \displaystyle \prod \limits_{\alpha \in A} X_\alpha Lindelof is to make some of the factors compact such that the product of the remaining non-compact factors is Lindelof. Then the product space is essentially the product of a compact space and a Lindelof space, which is always Lindelof.

For example, let X_0=\mathbb{R} and let X_\alpha=[0,1] for all \alpha with 0<\alpha<\omega_1. Then the product space \displaystyle \prod \limits_{\alpha \in A} X_\alpha is Lindelof since it is essentially the product of a compact space and a Lindelof space. However, the product \displaystyle \prod \limits_{\alpha \in A} X_\alpha is not hereditarily Lindelof.

In fact, the product space in the previous paragraph is \sigma-compact (i.e. the union of countably many compact sets). To make the example not \sigma-compact, simply make the first factor space a non-locally compact Lindelof space. For example, use the Sorgenfrey line or the space of the irrational numbers.

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