#Is the Weierstrass function a manifold?

9 messages · Page 1 of 1 (latest)

cedar ruin
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Ever since I learned about manifold I always had in mind if this (the graph of the Weierstrass function) is a 1 dimensional manifold (not necessarly a differentiable one) but I never had the time to think about it ( I mean it is a line segment basically at the end of the day, just jagged) so if anyone can help me with this (reasoning with it while talking would be better).

Weierstrass_Animation.gif
wanton iceBOT
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cedar ruin
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Let me remember the definition of a manifold:
A topological space M is a manifold if :

  1. Has countable base
  2. Hausdorff
  3. Every point x in M has a neighborhood Vx and a chart function phi:Vx->R such that it is a homeomorphism
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But I don't know if the graph of the Weierstrass function ( I will just call it Weier for shorthand) is a topological space in the first place. I don't remember that much from my analysis class so I am rusty on anything topological

lunar gorge
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the graph of any continuous function between euclidean spaces is

lunar gorge
cedar ruin
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+close