# 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.