Specific question:
"Suppose $X$ is a three dimensional Euclidean space with the standard Euclidean metric. Let $Y$ be the subset defined by $Y=\{P_1$ s.t. $P_1=(a_1,b_1,c_1)$ and $c_1=0\}$ and use the subspace topology for Y. Define a mapping $S:Y\to X$ as $S(a,b,0)=(a,b,+\sqrt{a^2+b^2})$ and define the subset of X by $Z=S(Y)$.
Prove that for every $(a,b)$ in $R^3$, the intersection Z with open balls at center $(a,b,+\sqrt{a^2+b^2})$ and of arbitrary dimension $r$ generates a basis for topology of Z."
My question doesn't specifically want an answer to the above question. rather, I ask for an explanation of the general concepts at play here. How can I go about solving these types of problems, other than saying that it is true by definition (which I think is incorrect). My misunderstanding is more fundamental than situational, so really any help would be greatly appreciated!
The basic concept is that you want to show two things namely; 1) Given some arbitrary open set, you can express that open set as a union of basis elements. 2) Show that if an open set is contained in two basis elements, say B1 and B2, then there is a basis element B3 contained in the intersection of B1 and B2 which also contains that open set.
Showing that (1) and (2) are satisfied by a collection of open sets is sufficient for proving that collection is a basis that generates the topology in question.