##  [Loop Space](/loop-space) 

  ##  [Loop Space](https://puremath.quantumdictionary.io/loop-space-0) 

  

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Definition

Given a pointed topological space (X, x0), the loop space ΩX is the space of continuous maps γ: [0,1] → X with γ(0) = γ(1) = x0 (equivalently based maps S^1 → X) equipped with the compact-open (or function) topology. Points are based loops and the space is typically pointed by the constant loop.