What is the relation of basis in linear algebra and basis in topology?

Question

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.

Answer 1

Well, they are two different entities: they are similar in that they are minimal information that give rise to a mathematical object--vector space in linear algebra, a topological space in General topology.

To see that they are different too -- I have adopted the following discussion from Munkres' book on general topology:

Firstly, note the following lemma:

Lemma (LA)

Let $V$ be a vector space and let $\scr B$ be a basis for $V$. Then, every vector $v \in V$ is uniquely written as a linear combination of basis elements.

Lemma (GT)

Let $(X, \scr T)$ be a topological space. Let $\scr B$ be a basis for topology $\scr T$ on $X$. Then, every open set in $\scr T$ can be expressed as a union of basis elements.

The difference arises from the fact that, this expression for the union need not be unique.

Let's look at a situation where this is actually the case:

Consider the three element set, $X=\{a,b,c\}$ and the following subset of $2^X$:

$$B=\{\{a\},\{a,b\}, \{a,b,c\}\}$$

It is not hard to see that $B$ is in fact a basis for some topology, $\scr T$ on $X$. Now, observe that $\scr T$ is actually:

$$\mathscr{T}=\{\varnothing, X, \{a\}, \{a,b\} \}$$

Now see that $$\{a,b,c\}=\{a,b,c\}=\{a,b\} \cup \{a,b,c\}=\{a\}\cup\{a,b\} \cup \{a,b,c\}$$

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I'll add two more points from the discussion I had with the user t.b.:

1. Topological bases are a special kind of small collection of open subsets on which you have a nice handle. They help study all open sets. Now, notice that there is no minimality in question nor independence, which however fails to make much sense.

2. The importance of bases in Linear algebra comes from the fact they are minimal spanning sets and maximal linearly independent sets. Notice that there are not immediate analogues of these facts for topological bases.

Answer 2

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".