📚 Usual Topology on ℝ | Complete Proof | Topology Series | The Math Professor

📚 Usual Topology on ℝ | Complete Proof | Topology Series | The Math Professor 🔥 How do we prove that the collection of usual open sets on \(\mathbb{R}\) actually forms a topology? In this lecture, we study the usual (standard) topology on \(\mathbb{R}\) and prove it rigorously from the definition of an open set. 📖 Topics Covered: ✅ Usual / Standard Topology on \(\mathbb{R}\) ✅ Open Intervals in \(\mathbb{R}\) ✅ Definition of an Open Set ✅ \(\varepsilon\)-Neighbourhood of a Point ✅ Construction of the Usual Topology ✅ Verification of the Three Topology Axioms ✅ Proof that \(\varnothing\) and \(\mathbb{R}\) Are Open ✅ Arbitrary Union of Open Sets Is Open ✅ Finite Intersection of Open Sets Is Open ✅ Detailed Step-by-Step Proof ✅ Important Examples & Observations 🎯 Key Idea: A subset \(U\subseteq\mathbb{R}\) is open in the usual topology if for every \(x\in U\), there exists some \(\varepsilon greater than 0\) such that $$ (x-\varepsilon,x+\varepsilon)\subseteq U. $$ 🔥 Main Journey: OPEN INTERVALS → OPEN SETS → USUAL TOPOLOGY → TOPOLOGY AXIOMS → COMPLETE PROOF The emphasis is on understanding why the axioms hold, so that the same proof technique can later be applied to other topological spaces. 🎯 Useful for HPPSC Assistant Professor Mathematics, M.Sc./B.Sc. Mathematics, CSIR-UGC NET, GATE, SET and other higher mathematics examinations. 📌 TOPOLOGY SERIES | USUAL TOPOLOGY ON ℝ | PROOF 👍 Like • Share • Subscribe to The Math Professor Learn • Understand • Excel #Topology #UsualTopology #StandardTopology #OpenSets #OpenIntervals #TopologicalSpace #TopologyProof #HPPSC #CSIRNET #GATE #SET #TheMathProfessor