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loop space (Definition)

Let $ X$ be a topological space, and give the space of continuous maps $ [0,1]\to X$, the compact-open topology, that is a subbasis for the topology is the collection of sets $ \{\sigma : \sigma(K)\subset U \}$ for $ K\subset [0,1]$ compact and $ U\subset X$ open.

Then for $ x\in X$, let $ \Omega_{x}X$ be the subset of loops based at $ x$ (that is $ \sigma$ such that $ \sigma(0) = \sigma(1) = x$), with the relative topology.

$ \Omega_{x}X$ is called the loop space of $ X$ at $ x$.



"loop space" is owned by mathcam. [ full author list (2) | owner history (2) ]
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See Also: suspension, Eilenberg-MacLane space
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Cross-references: relative topology, loops, subset, open, compact, collection, subbasis, compact-open topology, continuous maps, topological space
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This is version 4 of loop space, born on 2002-02-02, modified 2003-07-25.
Object id is 1640, canonical name is LoopSpace.
Accessed 3456 times total.

Classification:
AMS MSC: 54-00 (General topology :: General reference works )
Pending Errata and Addenda
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