# End-point-preserving homotopic paths

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Not important, terminology is from Mond's 2013 lecture notes. Some authors use path homotopy but we reserve this for map homotopy where the maps are paths (ie a free homotopy), we use this to distinguish homotopic paths ([ilmath]\text{rel }\{0,1\} [/ilmath]) from just paths that are homotopic maps Alec (talk) 14:31, 25 April 2017 (UTC)

## Definition

Let [ilmath]p,g:[0,1]\rightarrow X[/ilmath] be paths in a topological space [ilmath](X,\mathcal{ J })[/ilmath], we say they are end-point-preseriving-homotopic - or epph for short - if they are homotopic maps relative to [ilmath]\{0,1\} [/ilmath]