How does a topology establish a structure on a given set? Let us consider $\mathbb R^2:=\text{{(m,n)| m $\in$ $\mathbb R$ $\land$ n $\in$ $\mathbb R$}}$. 
As a set $\mathbb R^2$ is just a "powder". We can not yet talk about a plane, because nothing prevents us from "cutting" it until we get a "powder". 
Things are different for the topological space $\mathbb ({\mathbb R^2},\mathcal{T})$, where $\mathcal{T}$ is the standard topology on $\mathbb R^2$. It can only be "bended" or "stretched" - no "cutting". 
I am asking myself how a topology, just another set, can establish a structure such that no "cutting" is possible. My intuition tells me that we need a map and of course 2 topologies in order to talk about continuity, which somehow does this job.
Please do not restrict yourself to $\mathbb R^d$ within your argumentation. 
 A: Intuitively you can think of the way that the topology holds $\mathbb{R}^2$ together as coming from limits. A topology is enough structure to say which sequences in $\mathbb{R}^2$ converge and to what, despite being much less structure than a metric which is usually the first structure you meet that tells you about convergence. While a metric gives you a precise notion of "close" (two points being close together), what a topology gives you is a notion of "eventually arbitrarily close," which is good enough to form limits.
Once you have limits, you have the basic ingredients to understand why $\mathbb{R}^2$ cannot be "cut" - more specifically, the claim is that it is connected, which means it cannot be cut up into two disjoint pieces both of which are closed under limits. (This definition does not work in general but it's fine here.) Intuitively, the idea is that any way of chopping $\mathbb{R}^2$ up into two pieces introduces a "boundary" between the two pieces, and you can find sequences in each of the two pieces which converge to points on the boundary, and hence which must belong to both pieces; but this contradicts the assumption that the pieces are disjoint. 
