Real Analysis Lecture-28 | Separated Sets, Connected & Disconnected Sets

In this lecture, we cover one of the most important topics in Real Analysis: Connectedness and Separated Sets in ℝ. This session is highly useful for GATE 2026, IIT-JAM, CSIR-NET, and MSc Mathematics preparation. We start with the definition of separated sets, explore their properties, and understand the key difference between disjoint and separated sets. Then we move to connected and disconnected sets, including important results like: ✔ Intervals are connected ✔ Connected subsets of ℝ are exactly intervals ✔ Only ∅ and ℝ are both open and closed (clopen sets) ✔ ℚ (rationals) is NOT connected This lecture includes theory + proofs + examples, making it perfect for conceptual clarity and exam preparation. ⏱️ TIMESTAMPS 00:00 Introduction 01:30 Separated Sets – Definition 05:00 Examples of Separated Sets 08:00 Properties (Disjoint vs Separated) 12:00 Open Sets & Separation 15:00 Closed Sets & Separation 18:00 Disconnected Sets 22:00 Connected Sets 25:00 Intervals are Connected 29:00 Connected Subsets of ℝ ⇔ Intervals 32:00 Clopen Sets Theorem 34:00 ℚ is Not Connected + Summary 📚 Topics Covered Real Analysis, Connected Sets, Separated Sets, Disconnected Sets, Intervals, Topology on ℝ, Open and Closed Sets, Clopen Sets, Rational Numbers 🎯 Useful For GATE 2026 Mathematics IIT JAM Mathematics CSIR NET Mathematical Sciences BSc / MSc Mathematics Students 📺 Subscribe for Daily Maths Lectures 👉 https://youtube.com/@gim2932 #RealAnalysis #ConnectedSets #GATE2026 #IITJAM #CSIRNET #Mathematics #Topology #Intervals #ClopenSets #MathLecture 🚀