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I am confused with the concept of topology base. Which are the properties a base has to have?

Having the next two examples for $X=\{a,b,c\}$:

1) $(X,\mathcal{T})$ is a topological space where $\mathcal{T}=\{\emptyset,X,\{a\},\{b\},\{a,b\}\}$. Which is the general procedure to follow in order to get a base for $(X,\mathcal{T})$? Can $\mathcal{B}=\{\{a\},\{b\}\}$ be a base?

2) Having just $X$ and no topology $\mathcal{T}$ defined for $X$, is $\mathcal{A}=\{X,\{a\},\{c\}\}$ a base? What is the topology that it generates?

Thank you very much.

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2 Answers

up vote 3 down vote accepted

Let $X$ be a non-empty set. A collection $\mathscr{B}$ of subsets of $X$ is a base for some topology on $X$ if it satisfies two conditions:

  1. $\mathscr{B}$ covers $X$. That is, every point of $X$ belongs to at least one member of $\mathscr{B}$.
  2. If $B_1,B_2\in\mathscr{B}$ and $x\in B_1\cap B_2$, then there is a $B_3\in\mathscr{B}$ such that $x\in B_3\subseteq B_1\cap B_2$.

These two conditions are exactly what is needed to ensure that

$$\mathscr{T}=\Big\{\bigcup\mathscr{A}:\mathscr{A}\subseteq\mathscr{B}\Big\}$$

is a topology on $X$. In words, the collection of all unions of members of $\mathscr{B}$ is a topology on $X$, the topology generated by the base.

Note that a topology may have many different bases. The topology $\big\{\varnothing,\{a\},\{b\},\{a,b\}\big\}$ on the set $\{a,b\}$ has the following bases:

  1. $\big\{\varnothing,\{a\},\{b\},\{a,b\}\big\}$
  2. $\big\{\{a\},\{b\},\{a,b\}\big\}$
  3. $\big\{\varnothing,\{a\},\{b\}\big\}$
  4. $\big\{\{a\},\{b\}\big\}$

Conditions (1) and (2) above are the easiest way to characterize the families of sets that are bases for some topology on $X$. If you already have a topology $\mathscr{T}$ on $X$, you can say simply that a subset $\mathscr{B}$ of $\mathscr{T}$ is a base for $\mathscr{T}$ if and only if every member of $\mathscr{T}$ (i.e., every open set in the space $\langle X,\mathscr{T}\rangle$) is a union of members of $\mathscr{B}$.


Yes, $\big\{\varnothing,\{a\},\{c\},X\big\}$ is a base for a topology on $X$: it satisfies the two conditions given at the beginning of this answer. The topology that it generates is

$$\big\{\varnothing,\{a\},\{c\},\{a,c\},\{a,b,c\}\big\}\;.$$

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The third base that you propose, {{a},{b}}, should not be {0,{a},{b}}? Because {a}Intersection{b} = 0. And 0 is not in {{a},{b}}.Yes? –  Haritz Nov 20 '12 at 10:50
    
@Haritz: No: it is never necessary to include $\varnothing$ in a base. Condition (2) says that if there is a point $x\in B_1\cap B_2$, then there must be a $B_3\in\mathscr{B}$ such that $x\in B_3\subseteq B_1\cap B_2$; it says nothing about the case in which $B_1\cap B_2=\varnothing$. –  Brian M. Scott Nov 20 '12 at 10:56
    
Thank you very much @Brian M. Scott –  Haritz Nov 20 '12 at 11:01
    
@Haritz: You’re very welcome. –  Brian M. Scott Nov 20 '12 at 11:02
    
So {0,{a},{b}} will also be a base i guess? –  Haritz Nov 20 '12 at 11:05
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A base is a collection $B$ of sets such that every set in the topology can be written as a union of sets in $B$.

1) Can you write every set in $T$ as a union of $\{a\}$ and/or $\{b\}$?

2) Produce all the unions then you'll see what the topology is.

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Nice answer! "every set in the topology" or "every open set in the topology"? –  Tim Dec 31 '12 at 23:49
    
@Tim Both: every set in the topology is open. And on the other hand only sets in the topology are open. –  Matt N. Jan 1 '13 at 7:29
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