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Topological Space
★★★★☆Undergraduate
📖Definition
A topological space is a structure consisting of a set and a collection of open sets (topology). It allows defining continuity, convergence, and connectedness.
📐Formulas
∅, X ∈ τ
Empty set and whole set are open
\bigcup_α U_α ∈ τ
Arbitrary union of open sets is open
U₁ ∩ U₂ ∈ τ
Finite intersection of open sets is open
✏️Examples
예제 1
Explain why open intervals are open sets in ℝ.
📜History
Discovered by: Felix Hausdorff (1914)
Hausdorff gave the axiomatic definition of topological spaces.
⚡Applications
Analysis
Generalization of continuous functions
Algebraic Geometry
Definition of manifolds
Quantum Mechanics
Hilbert spaces
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