I am going through the basics of topology, in order to deal with topological vector space. I haven't taken any course of topology so I have some fundamental questions (I 've seen only some topological notion and definitions in other courses) Well, let $X$ be a topological vector space and $x\in\ X$
A neighbourhood $V$ of a point $x$ is always consider to be an open set and why? I know that neighbourhoods are defined as sets containing an open set that contains $x$ but I can't figure out how a neighbourhood is an open set by itself. Same question about the basis of neighbourhoods of $x$
We know that if $A\subseteq X$ an open set then $co(A)$ is also an open set (where $co(A)$ is the convex case of $A$) How can someone conclude from the above that $co(A)$ is a neighbourhood of $0$ ?
Is the following right or have I misunderstood something ?
$"$ Let $a\in\ K $ where $K$=($\Bbb R$ or $\Bbb C$) . Then we have $ax$ vector and $V$ a neighbourhood of $ax$ . There is neighbourhood $U$ of $x$ and $r \gt 0$ such as if $b\in\ K$ : $|b-a|\lt r$ and $y\in\ U$ such as $bx\in\ V$, more specific $bU\subseteq V$ $"$
If this sentence is true , could somebody explain it to me with simple words?
I would appreciate any help! Thanks in advance!!
Ad 1: A neighborhood can be defined to be always open, but as far as I know a neighborhood $N \subset X$ of $x \in X$ is usualy defined as a subset of $X$ containing some $U$ open with $x \in U$. $N$ doesn't need to be open. For instance $[0,1]$ is a neighborhood of the point $x = \frac{1}{2}$ as $ x\in (0,1) \subset[0,1]$.
Ad 2: For this to hold you need $0 \in co(A)$. In the general setting you gave it's not necessarily true:
Consider the open ball $B_1(3,3) = \{y \in \mathbb R^2 \mid \Vert y - (3,3) \Vert_2 < 1\}$. Obviously the origin is not in $B_1(3,3)$, but $co(B_1(3,3))=B_1(3,3)$ as the ball is convex itself.
So now assume $0 \in A$ for a moment: Since $A$ is open you know that $co(A)$ is open and hence $co(A)$ is a neighborhood of $0$ (as it contains itself and is an open set).
Ad 3: I have trouble understanding what it means. Is $K$ supposed to be the underlying field of the vector space? What are your assumptions, what the implications?