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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.

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They're conceptually similar in that both of them "generate" something (the vector space and the topology, respectively), neither is a special case of the other or anything like that. –  Alex Becker Apr 6 '12 at 7:13
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Both terms rely on the embedded/base language (English), whose cognate ('base') is thus overloaded. –  bgins Apr 6 '12 at 7:18
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I will second Alex Becker's comment. Two points: in topology there is little point in seeking a 'minimal' generating set. OTOH in linear algebra this is paramount given that we want to use the basis to introduce coordinates. Also, natural language is not rich enough to capture all the subtleties encountered in math. We just need to make the most of it. –  Jyrki Lahtonen Apr 6 '12 at 7:19
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They are unrelated, other than by accident of language. –  copper.hat Apr 6 '12 at 7:32
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The only important reason that the worse "base" is used to thoroughly in mathematics is to allow one mastering mathematics say "All your base are belong to us!". –  Asaf Karagila Apr 6 '12 at 12:13

2 Answers 2

up vote 3 down vote accepted

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


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.

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There is a sort of minimality notion for topological bases, in that for a given topological space X there is a unique least possible cardinality of a base of X. (This is known as the weight of X.) But not every base has this cardinality, e.g. the collection of all open sets of X is itself a base. And one base can strictly contain another base, unlike the linear-algebraic notion. –  Brad Apr 6 '12 at 15:09

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".

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