My name is Raheel Ahmad and I'm a qualified Mathematician, serving as a Lecturer at Govt. College University Faisalabad, Chiniot Campus (GCUFC). The main purpose of uploading video lectures is to share the knowledge and experience to other students and teachers and specially for those students who can't afford the expensive institutions. Every single video on my channel is made with Love and Hard work, So don't forget to leave a Like. and subscribe to my channel.
the given below class of the open-closed (clopen) intervals in the real line R is the base of the upper limit topology. B = { (a, b] : a, b in R, a < b } You can see, R (set of real numbers) is the union of the members of B because every real number belongs to some open closed interval. Similarly, the given below class of the closed-open (clopen) intervals in the real line R is the base of the lower limit topology. B = { [a, b) : a, b in R, a < b }. for further detail, find given below book with complete details. General Topology by Schaum's Outline Page # 88 Chapter # 6 Chapter Name: Bases and Sub-Bases
G bilkul ap thek keh rhy lekn hm ny un open sets k intersections ko consider krna ha jo Base k open sets ko generate krn aur ap delh skty ho k base ka hr open set generate ho rha ha to hm keh skty hn k S_1 is a sub-base for given topology.