I've studying the countability chapter in Schaum's General Topology and in the book, the author goes over first countable, second countable, and Lindelöf spaces. Once the author covers the definitions of each kind of space, he goes on to state two things:
- Proposition 9.2 A second countable space is also a first countable space
- Every second countable space is a Lindelöf space
So, when I first saw this, I thought that every Lindelöf space must therefore be first countable, but I know that's not true since $(X,\mathcal{D})$ where $X$ is an uncountable set and $\mathcal{D}$ is the discrete topology is first countable but not Lindelöf. So therefore, we can't state that every Lindelöf space is first countable.
In trying to understand the relationship between these three kinds of spaces, I made a diagram that I think nicely illustrates what I can gather about the relationship between these three. View Image
Once you click the link, you'll see I've made a Venn diagram illustrating what I think is the relationship between these three. In this way, from this illustration, we can gather two things:
Second countable spaces belong to both first countable spaces and Lindelöf spaces
There are examples of first countable spaces that are not Lindelöf and Lindelöf spaces which are not first countable.
Am I leaving out anything here or is this a good way to visualize the relationship between first countable, second countable, and Lindelöf?