Is the Sorgenfrey line separable?
The rational numbers are a dense subset of this topological space, so the Sorgenfrey line is separable.
Is Sorgenfrey a compact line?
We observe also that the Sorgenfrey line cannot be compact (since the usual topology on R is coarser and not compact). However, the Sorgenfrey line is hereditarily Lindelöf, i.e. every subspace is Lindelöf (Exercise).
Is lower limit topology compact?
The collection of all singletons of the space is an open cover which admits no finite subcover. Finite discrete spaces are compact. In ℝ carrying the lower limit topology, no uncountable set is compact. In the cocountable topology on an uncountable set, no infinite set is compact.
Is the real line normal?
The real line is a locally compact space and a paracompact space, as well as second-countable and normal. It is also path-connected, and is therefore connected as well, though it can be disconnected by removing any one point.
What do you mean by a regular space prove that a compact Hausdorff space is regular?
Theorem 4.7 Every compact Hausdorff space is normal. Proof. Let A and B be disjoint closed subsets of the compact Hausdorff space X. Then A and B are compact. Now use compactness of A to obtain open sets U and V so that A ⊂ U, B ⊂ V , and U ∩ V = 0.
How do you show 0 1 is not compact?
The open interval (0,1) is not compact because we can build a covering of the interval that doesn’t have a finite subcover. We can do that by looking at all intervals of the form (1/n,1).
Is the space RL connected?
One of the ways we characterize the connectedness of a space is that it is connected if and only if the only sets that are both open and closed are the sets X and ∅. To show that Rl is not connected, consider the set [0, 1). Rl = [0, 1) ∪ ((−∞, 0) ∪ [1, ∞)) and Rl is a union of disjoint, nonempty, open sets.