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We defined basis for a topology, and there is something that I do not understand. Here is how we defined the basis. Given a topological space $\left(X,\mathscr T\right)$ we defined basis for the topology to be the set $\mathscr B$ , consisting of subsets of $X$ if it satisfies 2 conditions.

First, for all $x\in X$ there exists $B\in\mathscr B$ such that $x\in B$ . Secondly if $x\in B_{1}\cap B_{2}$ , for $B_{1},B_{2}\in\mathscr B$ , then there exists $B_{3}$ such that $x\in B_{3}\subseteq B_{1}\cap B_{2}$ .

So,I am studying from the book Topology, by Munkres. And it stated that the basis is a subset of the topology. But, if I choose $X=\left\{ a,b\right\}$ and $\tau=\left\{ \emptyset,X\right\}$ , and I can define $\mathbb{B}=\left\{ \left\{ a\right\} ,\left\{ b\right\} \right\} $ . The set $\mathbb{B}$ satisfies the conditions of the definitions. However it's not a subset of the topology. What am I missing here?

Charles Carmichael
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4 Answers4

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First, I guess you mean $\mathscr T= \{\varnothing,X\}$.

Second, you should generally speak of "a" basis for a topology, rather than "the" basis for a topology. Given a topology, there are usually many distinct bases for this same topology.

Third, your set $\mathscr B = \{\{a\},\{b\}\}$ is indeed a basis for a topology, but not a basis for the topology you specified. Any collection of sets that is a basis will generate a topology, but every basis need not generate the same topology.

You probably want to specify that each of the sets $B\in\mathscr B$ satisfy $B\in\mathscr T$, i.e., that $\mathscr B\subset\mathscr T$.

MPW
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  • Yes, I meant the trivial topology. Now I understand what you mean. So in the definitions we have to add that every element of the base is an element of the given topology as well, right? Because, if the definition doesn't say that, then the example that I give is a basis for the given topology. – Charles Carmichael Mar 11 '15 at 16:58
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$\{\emptyset,\mathscr T\}$ is not a topology. It's a set containing the topology as an element.

Tim Raczkowski
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The definition of a basis is a collection of subsets of a nonempty set X, and this collection must satisfy the two conditions you specified. Then we can define the topology generated by this basis as arbitrary unions of sets in the basis, plus the empty set. Then we check that the collections of set thus generated is indeed a topology on the original set X. Then, of course, we see that the basis is a subset of the topology generated by it.

Conversely, we can first have a topological space (X, O) where O is a topology on the nonempty set X, and find a basis that generates the given topology O. We can also then readily check that this basis must be a subset of the given topology.

As you can see, the concepts of topology and basis can be induced from each other, but the fact that the basis is a subset of the topology it generates is a byproduct of the definition, not part of the definition.

James Harden
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If we already have a topology given on a set, then a basis of the topological space must be a subset of the topology given on the set and it must generate ( i.e. by taking arbitrary unions, finite intersections of the basis elements) the whole topological space. So, if you choose any topology on a set and try to find a basis of it, you need to keep in mind that the subset of the topology you choose must satisfy the conditions you mentioned as the definition of a basis from Munkres and this subset must generate the entire topology.

Now, if we have X just as a set and we take a collection of its subsets such that they satisfy the definition you mentioned of a basis, then this subset generates a topology on X which might be different from any other topology on X generated by some other basis.

Anusha Bhattacharya
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