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[parent] conditions for a collection of subsets to be a basis for some topology (Proof)

Not just any collection of subsets of $ X$ can be a basis for a topology on $ X$. For instance, if we took $ \mathcal{C}$ to be all open intervals of length $ 1$ in $ \mathbb{R}$, $ \mathcal{C}$ isn't the basis for any topology on $ \mathbb{R}$: $ (0,1)$ and $ (.5, 1.5)$ are unions of elements of $ \mathcal{C}$, but their intersection $ (.5,1)$ is not. The collection formed by arbitrary unions of members of $ \mathcal{C}$ isn't closed under finite intersections and isn't a topology.

We'd like to know which collections $ \mathcal{B}$ of subsets of $ X$ could be the basis for some topology on $ X$. Here's the result:

Theorem 1   A collection $ \mathcal{B}$ of subsets of $ X$ is a basis for some topology on $ X$ if and only if:
  1. Every $ x\in X$ is contained in some $ B_x\in \mathcal{B}$, and
  2. If $ B_1$ and $ B_2$ are two elements of $ \mathcal{B}$ containing $ x\in X$, then there's a third element $ B_3$ of $ \mathcal{B}$ such that $ x\in B_3\subset B_1\cap B_2$.
Proof. First, we'll show that if $ \mathcal{B}$ is the basis for some topology $ \mathcal{T}$ on $ X$, then it satisfies the two conditions listed.

$ \mathcal{T}$ is a topology on $ X$, so $ X\in \mathcal{T}$. Since $ \mathcal{B}$ is a basis for $ \mathcal{T}$, that means $ X$ can be written as a union of members of $ \mathcal{B}$: since every $ x\in X$ is in this union, every $ x\in X$ is contained in some member of $ \mathcal{B}$. That takes care of the first condition.

For the second condition: if $ B_1$ and $ B_2$ are elements of $ \mathcal{B}$, they're also in $ \mathcal{T}$. $ \mathcal{T}$ is closed under intersection, so $ B_1\cap B_2$ is open in $ \mathcal{T}$. Then $ B_1\cap B_2$ can be written as a union of members of $ \mathcal{B}$, and any $ x\in B_1\cap B_2$ is contained by some basis element in this union.

Second, we'll show that if a collection $ \mathcal{B}$ of subsets of $ X$ satisfies the two conditions, then the collection $ \mathcal{T}$ of unions of members of $ \mathcal{B}$ is a topology on $ X$.

  • $ \emptyset \in \mathcal{T}$: $ \emptyset$ is the null union of zero elements of $ \mathcal{B}$.
  • $ X\in \mathcal{T}$: by the first condition, every $ X$ is contained in some member of $ \mathcal{B}$. The union of all the members of $ \mathcal{B}$ is then all of $ X$.
  • $ \mathcal{T}$ is closed under arbitrary unions: Say we have a union of sets $ T_{\alpha}\in \mathcal{T}$...
    $\displaystyle \bigcup_{\alpha \in I} T_{\alpha}$ $\displaystyle = \bigcup_{\alpha \in I} \bigcup_{\beta \in J_{\alpha}} B_{\beta}$    

    (since each $ T_{\alpha}$ is a union of sets in $ \mathcal{B}$)


      $\displaystyle = \bigcup_{\beta \in \bigcup_{\alpha \in I} J_{\alpha}} B_{\beta}$    

    Since that's a union of elements of $ \mathcal{B}$, it's also a member of $ \mathcal{T}$.

  • $ \mathcal{T}$ is closed under finite intersections: since a collection of sets is closed under finite intersections if and only if it is closed under pairwise intersections, we need only check that the intersection of two members $ T_1, T_2$ of $ \mathcal{T}$ is in $ \mathcal{T}$.

    Any $ x\in T_1\cap T_2$ is contained in some $ B_x^1\subset T_1$ and $ B_x^2\subset T_2$. By the second condition, $ x\in B_x^1\cap B_x^2$ gets us a $ B_x^3$ with $ x\in B_x^3 \subset B_x^1\cap B_x^2 \subset T_1\cap T_2$. Then

    $\displaystyle T_1\cap T_2 = \bigcup_{x\in T_1\cap T_2} B_x^3 $

    which is in $ \mathcal{T}$.

$ \qedsymbol$



"conditions for a collection of subsets to be a basis for some topology" is owned by waj.
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Keywords:  teaching proofs, characterization of a basis, what a basis looks like

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Cross-references: zero elements, null, open, contained, finite, closed under, intersection, unions, length, open intervals, topology, basis, subsets, collection
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This is version 1 of conditions for a collection of subsets to be a basis for some topology, born on 2004-05-10.
Object id is 5845, canonical name is ConditionsForACollectionOfSubsetsToBeABasisForSomeTopology.
Accessed 2498 times total.

Classification:
AMS MSC54A99 (General topology :: Generalities :: Miscellaneous)
 54D70 (General topology :: Fairly general properties :: Base properties)

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