##  [Path Space](/path-space) 

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

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

The path space PX of a topological space X is the space of continuous maps α: [0,1] → X, typically endowed with the compact-open topology. Variants include based path spaces (fixing one or both endpoints) and spaces of paths with specified endpoint conditions. PX is a function space whose points are parametrized paths in X.