Is $W$ connected or nowhere dense? 
Let,$W\subset \mathbb R^{n}$ ba a linear subspace of dimension at most $n-1$. Which of the following statement(s) is/are true?
(a) $W$ is  nowhere dense.
(b) $W$ is closed.
(c) $\mathbb R^{n}\setminus W$ is connected.
(d) $\mathbb R^{n}\setminus W$ is disconnected.

My Attempt:
Suppose that, $n=2$. Then $W\subset \mathbb R^{2}$. Further suppose that $W=\mathbb R$. Then $\mathbb R^{n}\setminus W$ is disconnected.
So, option (c) is false & option (d) is true.
Again, $Int(\bar W)=Int(W)=W\not =\phi $. So, $W$ is NOT nowhere dense.
So, option (a) is false.
I have no idea about option (b).
Check my arguments & help about option (b).
 A: I'll give you the response but you should try to find the proofs, as they are not difficult at all.


*

*In fact $W$ is nowhere dense (think about a neighborhood in $ℝ^n$ of a point of $W$, so what's the interior of $W$?)

*Of course $W$ is closed, it's a subspace (prove it)

*(and 4.) It depends on the dimension, if it's exactly $n-1$ then the complement is disconnected (think about a plane in $ℝ^3$), otherwise it is connected.

A: rewritten is right. Here are some tips to make this more rigorous:
First for the n-1 case.
Write $W=\mathrm{span}\{e_1,\ldots,e_{n-1}\}$. 
This can be extended to a basis $\{e_1,\ldots,e_n\}$ of $\mathbb{R}^n$.
Define $f(a_1e_1 + \cdots + a_ne_n)=a_n$. You can check that $f$ is linear and continuous. 
(a) Might be easiest to show the complement is open and dense. For open, see (b) first. For dense, any $a_1e_1+\cdots+a_{n-1}e_{n-1} + 0e_n$ can be approached from $W^C$ by keeping the $a_i$ the same except replacing $a_n$ by a sequence approaching 0.
(b) $W=f^{-1}(\{0\})$. Since $\{0\}$ is closed, $W$ is closed.
(c and d) Let $A=f^{(-1)}(\{x\in\mathbb{R}:x>0\}$ and $B=f^{(-1)}(\{x\in\mathbb{R}:x<0\}$. Then $A$ and $B$ are open and their union is $W^C$.
If the dimension of $W$ is less than $n-1$ then
(b) $W$ is the intersection of a number of $n-1$ dimensional subspaces.
(a) Same as above
(c and d) If its path connected then its connected. So construct a path. You have at least two dimensions to work with. For instance, you could paste three paths together to move to the opposite side of $W$: one going along $e_{n}$, another along $e_{n-1}$, and the last in the opposite direction from the first path along $e_{n}$. 
