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I am beginning to learn path connectivity (in the beginning of a complex analysis course) and am trying to sort out a few issues.

We defined a path to be a continuous map $\gamma:[a, b] \to \Bbb R^2$.

Firstly I want to convince myself that the image of such a path must "look like" a curve, or in other words $\gamma([a, b])$ must always look like a traversal drawn by a pen on the plane without lifting it.

But doing a bit of googling it seems there are weird pictures that arise as images of paths and curves. One of them being the Peano Curve but understanding it requires prerequisites I do not have. So,

Is there an example of a continuous function $\gamma:[a, b] \to \Bbb R^2$ whose image is fat and does not look like the traversal of a point on the plane, constructed using somewhat elementary methods?

Ishfaaq
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    The Peano curve is very much continuous. So, you may wish to rephrase your question. – Ittay Weiss Aug 13 '15 at 03:20
  • @IttayWeiss: May have gotten confused. I think I read somewhere that there is no continuous map from $[0,1]$ onto $[0,1] \times [0,1]$. Is that right? – Ishfaaq Aug 13 '15 at 03:25
  • The Koch snowflake is "fat" too, and the construction is also fairly elementary. – Silvia Ghinassi Aug 13 '15 at 03:26
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    @Ishfaaq Space-filling curves are continuous. In general, a curve is continuous by definition. – Silvia Ghinassi Aug 13 '15 at 03:27
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    @Ishfaaq you may have read that there is no continuous bijection $[0,1]\to [0,1]\times [0,1]$. – Ittay Weiss Aug 13 '15 at 03:29
  • The point is that the Peano curve passes through EVERY point in the square, although the point moves continuously as the parameter moves through the interval. ${}\qquad{}$ – Michael Hardy Aug 13 '15 at 03:31
  • @SilviaGhinassi: I was trying to read the Wikipedia Article on Space Filling Curves but like I said it is sadly too sophisticated for me. And if you will, why is the Koch Snowflake fat? – Ishfaaq Aug 13 '15 at 03:31
  • @Ishfaaq The intuitive concept of "fat" can be captured in mathematics by the Hausdorff dimension. Being "fat" is having Hausdorff dimension $>1$. In words, if you look at the construction, you can see that as the process iterates the curve gets "fatter", meaning that when you zoom in the curve is never a line, and it's infinitely detailed. – Silvia Ghinassi Aug 13 '15 at 03:38
  • Here are some ways of constructing a space filling curve (http://math.stackexchange.com/questions/141958/why-does-the-hilbert-curve-fill-the-whole-square). If you know Banachs fixed point theorem (also called the contraction principle), you are good to go. – PhoemueX Aug 13 '15 at 06:27

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