## trouble understanding the relative topology and subspaces

i am aware of the formal definition of relative topology where one lets (X,T) be a topological space and let S be a subset of X, and the relative topology is all {U intersect S: U is in T}. what is a good way to intuitively think of these sets? (i have a trouble visualizing them) also, can these sets form a subspace of a certain topological space? for example, in R^2 with its usual topology, how would i describe the topology on E when it is viewed as a subspace of R^2?