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1. • CommentRowNumber1.
   • CommentAuthorporton
   • CommentTimeOct 10th 2015
   • PermaLink
   Author: porton
   Format: MarkdownItexI need to ensure that this is not an error: [This page](http://ncatlab.org/nlab/show/directed+topological+space) says: > (reparameterization) $dX$ is closed under composition with increasing maps $I\to I$; So that these "increased maps" are continuous is not required?

   I need to ensure that this is not an error:

   This page says:

   (reparameterization) $dX$ is closed under composition with increasing maps $I\to I$;

   So that these “increased maps” are continuous is not required?

2. • CommentRowNumber2.
   • CommentAuthorporton
   • CommentTimeOct 10th 2015
   • (edited Oct 10th 2015)
   • PermaLink
   Author: porton
   Format: MarkdownItexConsider path $I\rightarrow I$ defined by the identity function $id_I$. Then compose it on the left with a non-continuous increasing function $I\rightarrow I$. Then the composition is not a path at all and thus cannot be a member of $dX$. So $dX$ cannot be closed under above mentioned composition. Where is the error?

   Consider path $I\rightarrow I$ defined by the identity function $id_I$.

   Then compose it on the left with a non-continuous increasing function $I\rightarrow I$. Then the composition is not a path at all and thus cannot be a member of $dX$. So $dX$ cannot be closed under above mentioned composition.

   Where is the error?

3. • CommentRowNumber3.
   • CommentAuthorDavidRoberts
   • CommentTimeOct 10th 2015
   • (edited Oct 10th 2015)
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItex_Maps_ in topology are by default continuous. It is mentioned when they are not, and sometimes the terminology reverts to 'functions' as an additional flag that something is going on. See [[map]], which should have been you first port of call.

   Maps in topology are by default continuous. It is mentioned when they are not, and sometimes the terminology reverts to ’functions’ as an additional flag that something is going on.

   See map, which should have been you first port of call.