Q:

lindelöf and second countable spaces

Can anyone give me some examples and non examples of lindelöf or second countable space and spaces that is lindelöf but not second countable?   And I understand the definition but find it is hard to visualize and imagine.
I have tried google it but it turns out I only found some silly examples like finite set or empty set.\
In general, how can one construct a topological space that is lindelöf or second countable?\
Someone in stack exchange said the real line with discrete topology is lindelöf, but I do not think so. We can simply construct an open cover defined by the collection of all the singleton set. And this open cover is well defined since singleton set is open in discrete topology. Hence, by definition it is not lindelöf.\
Last question, is (0,1) in the real line equipped with usual topology lindelöf? I think it is lindelöf but I could not give any formal proof.
(0,1) fails to be a compact set since we can construct an open cover defined by (1/n,1-1/n) but this open cover does not work so well for arguing for lindelöf property since quotient number is dense in (0,1). So intuitively I think it is lindelöf.\
I wrote a pretty long question. My mothertongue is not English. Hopefully, you guys can understand me.

A:

The Sorgenfrey line, also called the lower-limit topology on the real line. It has a basis of intervals $[a,b)$ (or some authors prefer $(a,b]$, upper-limit topology). 
It is hereditarily Lindelof (and hereditarily separable, i.e. every subspace has a countable dense subset), but it is not second countable. 
Real line with discrete topology is not Lindelof. 
$(0,1)$ with the usual topology is Lindelof, it is homeomorphic to the real line. 
The real line is $\sigma$-compact, that is union of countably many compact subspaces. The real line is the union $\cup_n[-n,n]$, clearly $\sigma$-compact. 
The interval $(0,1)=\cup_n [\frac1n,1-\frac1n]$ is $\sigma$-compact. 
Every $\sigma$-compact space is Lindelof (easy to prove). 
In particular, clearly every countable space is Lindelof. But not every countable space is second countable (even with nice separation axioms). For example, the countable sequential fan is not second countable. It is obtained by taking a disjoint family of countably many convergent sequences, and 
"gluing" their limit points into one limit point, via a quotient map. The result is not first countable at that point. (I couldn't find a suitable online reference to the countable sequential fan, but it has similar properties to the quotient space $\Bbb R/\Bbb N$, which is also not first countable, and likely discussed in most topology books.) 
There is an online searchable database (called $\pi$-base), you can make a query asking for Lindelof, not second countable spaces. For many more examples see 
https://topology.jdabbs.com/spaces?q=lindelof%20%2B%20~Second%20Countable
Every second countable space is Lindelof (you may need to assume some separation axioms, often included in the definitions). 
One of the examples at $\pi$-base is the one-point Lidelofication of uncountable discrete space. Take any uncountable set, and a point $p$, and isolate all point but $p$. The neighborhoods of $p$ are co-countable (that is, they have a countable complement). The definition easily implies that the result is a Lindelof space. But it is not first-countable at $p$, and hence not second countable. Often (in the case when "uncountable" is taken to be the first uncountable cardinal) this space is described as the set of all countable ordinals $\omega_1=\{\alpha:\alpha<\omega_1\}$ together with the first uncountable ordinal $\omega_1$, so $X=[0,\omega_1]=\{\alpha:\alpha\le\omega_1\}$, with all countable ordinals $\alpha<\omega_1$ isolated, and with basic neighborhoods of $\omega_1$ 
of the form $(\alpha,\omega_1]$, with $\alpha<\omega_1$. This is, in addition, an example of a Linearly Ordered Topological Space (LOTS), with only one non-isolated point. (It is a LOTS under a somewhat different order, one may insert a decreasing sequence in front of every limit ordinal.) 
The last (and some of the previous examples) are not second countable, because they are not even first countable (and, for the one-point Lindelofication the point $p$ is not even a $G_\delta$ point, that is, it is not the intersection of any countable family of open sets, and the space is not $\sigma$-compact). On the other hand the Sorgenfrey line is perfectly normal: Every closed set (and in particular every point) is the intersection of a countable family of open sets (the proof that the Sorgenfrey line is hereditarily Lindelof uses this). But it is not second countable, since for every basis, and for every $x$ there must be a basic element $B_x$ with $x\in B_x\subseteq[x,\infty)$ and clearly if $x\neq y$ then $B_x\neq B_y$. 
An example of a space that is separable but not second countable and not Lindelof is the Moore, or Niemytzki plane (also called tangent-disk space, usually available in topology texts). 
There are also compact spaces (which of course is stronger than Lindelof) that are not second countable. One such example closely related to the Sorgenfrey line is the Alexandrov double arrow space, also called split interval. 
Another is the Alexandroff double circle. Note that every second countable space in hereditarily Lindelof (since every subspace is second countable, and hence Lindelof). The Alexandroff double circle is compact (and hence Lindelof), but has an uncountable discrete subspace, which of course is not Lindelof. Hence the Alexandroff double circle is not second countable. 

