Specifically, every second-countable space is separable (has a countable dense subset) and Lindelöf (every open cover has a countable subcover).

Lower limit topology: | In |mathematics|, the |lower limit topology| or |right half-open interval topology| is a ... World Heritage Encyclopedia, the aggregation of the largest online encyclopedias available, and the most definitive collection ever assembled. For metric spaces, however, the Example: , the real line with the lower limit topology, is not metrizable. For metric spaces, however, the

$\mathbb{R}$ with the lower limit topology is not second-countable. In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set of real numbers; it is different from the standard topology on (generated by the open intervals) and has a number of interesting properties.It is the topology generated by the basis of all half-open intervals [a,b), where a and b are real numbers. Remark. But there are an uncountable number of real numbers. The reverse implications do not hold. X satisfies the Second Countability Axiom, or is second-countable. For if at least one of the . But is NOT second countable. In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set of real numbers; it is different from the standard topology on (generated by the open intervals) and has a number of interesting properties.It is the topology generated by the basis of all half-open intervals [a,b), where a and b are real numbers. For metric spaces, however, the properties of being second-countable, separable, and Lindelöf are all equivalent. Remark. It is the topology generated by the basis of all half-open interval s ["a","b"), where "a" and "b" are real numbers.. 12 $\begingroup$ I am trying to prove that $\mathbb{R}$ with the lower limit topology is not second-countable. The proof: Note that is separable as is a countable, dense subset as every open set of the form contains a point of .

The reverse implications do not hold. Sosecond-countable is more restrictive than first-countable. It's not enough for a space to be first countable, because ℝ with the lower limit topology is first countable and seperable but not second countable. For example, the lower limit topology on the real line is first-countable, separable, and Lindelöf, but not second-countable. Ofcourse, ifa space is second-countablethenit is first-countable. Ask Question Asked 5 years, 4 months ago. On the other hand, a seperable metric space is always second countable. Note. Active 4 years, 7 months ago. This is a metric space (hence first countable), but a basis of this space must have uncountably many points, so it’s not second countable. Specifically, every second-countable space is separable (has a countable dense subset) and Lindelöf (every open cover has a countable subcover).
0. For example, the lower limit topology on the real line is first-countable, separable, and Lindelöf, but not second-countable. In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set R of real numbers; it is different from the standard topology on R and has a number of interesting properties. {0,1}^ω. 3. Why isn't the Sorgenfrey line sigma-compact? $\mathbb{R}$ with the lower limit topology is not second-countable. Viewed 9k times 26. It is not true that separable first-countable spaces are second-countable; a counterexample is the real line equipped with the half-open or lower limit topology that has as basis the collection of half-open intervals [a, b) [a, b).