Although during the test there are just probably a couple or triple of questions concerning topology, for me it is the most enigmatic and interesting topic among others. Nevertheless, there are some questions that are still baffling me.

1. Let's consider the set of real numbers R and two topologies defined on it:

T1 - standard real line topology;

T2 - lower-limit topology.

Therefore, I see that:

T1 = {empty set, R, all intervals (a,b)};

T2 = {empty set, R, all intervals [a,b)}.

At the same time, it is considered that T2 is strictly finer than T1. That actually implies that T1<T2 which assumes that T2 contains all elements of T1. But T2 does not have intervals (a,b)? Does it?

2. Let's consider again the set R of real numbers and the lower-limit topology T2. Apparently, with this topology R is Hausdorff space, since every two distinct points can be separated by neighborhoods. There is a property that

Every singleton in Hausdorff space is closed.

At the same time, according to the definition of closed set as complement of open set, I can see that only closed set in T2 are just

empty set and R (they are actually both open and closed in every topology);

[a,b) (they are actually both open and closed in T2);

(-inf,a).

But I can't think of example where singleton could be a complement of open set in T2. Unless, as it was discussed in question 1, the intervals (a,b) are also open in T2.

What you think about that?