First countable not Lindelöf space It is well known that any second-countable space is a Lindelöf space, but not conversely (see here). Moreover, any second-countable space is, clearly, first-countable.

First countable not Lindelöf space It is well known that any second-countable space is a Lindelöf space, but not conversely (see here). Moreover, any second-countable space is, clearly, first-countable. > Question. Does there exist a first countable space with is not Lindelöf?

The discrete topology provides an example on any non-countable set $X$.

It is clearly first-countable since $\\{x\\}$ is a local basis for any point $x$.

The open cover $\\{ \\{x\\} | x\in X\\}$ has no countable subcover.

First countable not Lindelöf space It is well known that any second-countable space is a Lindelöf space, but not conversely (see here). Moreover, any second-countable space is, clearly, first-countable. > **Question.** Does there exist a first countable space with is not Lindelöf?

First countable not Lindelöf space It is well known that any second-countable space is a Lindelöf space, but not conversely (see here). Moreover, any second-countable space is, clearly, first-countable. > Question. Does there exist a first countable space with is not Lindelöf?