Open Sets Definition I understand that the definition of an open set in a Metric Space and this can transfer over if you're dealing with a Metric Topological Space. However, I'm not sure if there is a standard definition of an open set in a Topological Space. I have read that it may vary under constraints.
I found one definition on Wikipedia:
If ($X$, $D$) is a Topological Space, then a subset $O$ of $X$ is open if and only if $O$ is a neighbourhood of each of its points.
This is how I expressed it mathematically:
for all x $\in$ O there exist an open set U containing x such that U  $\subseteq\ O $
 A: In General Topology there is no definition of the open set. It is just an element of the topology. 
If $(X,\tau)$ is a topological space, then $U\subset X$ is open iff $U\in\tau$.
A: There are different ways to define a general topological space.
One way is to list the neighborhoods for each of the points (so the neighborhoods are part of the definition of the topology). This is done in the book Elements of Point Set Topology by Baum. In this case an open set is defined to be a set which is a neighborhood of each of its points.
Another way to define a topology is to list all the open sets (so the open sets are part of the definition of the topology). This is the more typical way that I have seen it done. A book that does it this way is Introduction to Topology by Gamelin and Greene. In this case a neighborhood of a point $x$ is defined to be a set $U$ such that there is an open set $O \subset U$ with $x \in O$.
The two definitions of a topological space are equivalent, which I believe is proved at some point in Elements of Point Set Topology.
So the initial definition of open set depends on your definition of topological space (but they are equivalent). In one case the open sets are listed in order to define the space. In the other they are defined essentially by the definition that you supplied.
A: The definition is equivalent to the usual definition in a topological space. When you have a metric space $(X,\rho)$, you can consider the topological space $(X,\tau_\rho)$ where the topology $\tau_\rho$ consists of all sets in $X$ that are unions of open balls $B(x,\varepsilon)$. Observe that
$(1)$ $X$ is a union of open balls, namely $X=\bigcup_{x\in X} B(x,1)$, so $X\in \tau_\rho$.
$(2)$ $\varnothing$ is the empty union of balls, so $\varnothing \in \tau_p$.
$(3)$ The arbitrary union of sets that are union of balls is clearly a union of balls.
$(4)$ If $B(x,\varepsilon_1)$ and $B(y,\varepsilon_2)$ have nonempty intersection, we can always find a point $z$ and a ball $B(z,\varepsilon_3)$ contained in $B(x,\varepsilon_1)\cap B(y,\varepsilon_2)$. This can be used to prove the intersection of two sets that are union of balls is again a union of balls, hence is in  $\tau_\rho$ again.
Thus the tentative topology $\tau_\rho$ that $\rho$ induces is indeed a topology.
When dealing with metric spaces, we say that $O$ is a nbhd of $x$ if it contains an open ball centered at $x$. It can be shown open balls are nbhds of each of its points. Hence, open balls are open using the definition you supply. 
Now, we'd like to prove

PROP A set in a metric space $(X,\rho)$ is open (i.e. it is a nbhd of each of its points) iff it is the union of open balls, that is, iff it is in $\tau_\rho$.

PROOF Suppose the set $S$ is a union of open balls. Pick $x$ in your set. Since $S$ is a union of balls, $x$ must be in some ball in the union, call it $B_1$. But $B_1$ is a nbhd of each of its points, so it contains some open ball containing $x$. This ball will be contained in $S$, so $S$ is open. Conversely, suppose $S$ is a nbhd of each of its points. Then for each $x$ we can find a ball $B(x,\varepsilon_x)$ contained in $S$. Then $S=\bigcup_{x\in S} B(x,\varepsilon_x)$ will be a union of balls $\blacktriangleleft$. 
A: Given a topology $\tau$ on a set $X$, a set $\mathcal{O}$ is called open if and only if $\mathcal{O} \in \tau$. A topology is a collection of subsets of $X$ such that it contains $X$, is closed under all unions, and closed under finite intersections. This definition is consistent with the metric space definition of open.
The Wikipedia article on open sets gives a more in-depth explanation. The general topology article is another good place to start.
