TITLE: What is the relation of basis in linear algebra and basis in topology?
QUESTION [5 upvotes]: In linear algebra and topology ,it all has the concept basis,but I can not construct the relation of them,could you explain the relation of two basis,such as
the basis in linear algebra is special case of basis in topology,and some details about it,thanks a lot.

REPLY [2 votes]: As previously discussed, the two notions are not perfectly analogous, but here's one way in which they are conceptually similar. If you have a vector space $V$ and a basis $B$ for $V$, and you want to construct some linear-algebraic thing "living on $V$", it is equivalent to constructing something "living on $B$". For example, if you have another vector space $W$, then constructing a linear map $T:V\to W$ is equivalent to choosing a function $T|_B:B\to W,$ as each such function has a unique linear extension.
Similarly, if $X$ is a topological space and $B$ is a base for $X$, some things "living on $X$" can be equivalently defined as "living on $B$". For example, to construct a sheaf of abelian groups on $X$, it is equivalent to define a $B$-sheaf - an assignment of groups to elements of $B$ which satisfies the sheaf axioms "on $B$". (See the definition on Wikipedia.) Any $B$-sheaf extends uniquely to a sheaf on $X$, and conversely any sheaf on $X$ restricts to give a $B$-sheaf. (The same is true for sheaves with values in many other categories.)
In this sense, each notion of basis is "a (possibly) simpler gadget on which to define things".