# Two problems on topological properties of two different sets

Which properties hold for the following sets?
Open, connected, closed, nowhere dense?

1. $$A =\{(x,y) \in\mathbb{R}^2 \mid y=mx\}\setminus \{(0,0)\}\subset\mathbb{R}^2$$

2. $A$ is the closure in $C[0,1]$ of the set $B$ where
$$B:=\{f\in C^1[0,1]: |f(x)|\leqslant 1\mbox{ and }|f’(x)|\leqslant 1,\, \forall x\in[0,1]\}.$$

For (1) the straight line but origin missing. So it only nowhere dense. Am I right?

For (2) I have no idea. also guide me for few resoureces where I can find the different topological properties of the given set.

Thanks for your help.

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For (2), you need to use Ascoli-Arzelà theorem. (1) is correct. –  Giuseppe Negro Jan 4 '13 at 13:06
will you explain please –  poton Jan 4 '13 at 13:25
You should be clear about what you need explained. –  Chris Eagle Jan 4 '13 at 16:26
There is not much to be said that is not written here. –  Giuseppe Negro Jan 4 '13 at 17:23