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mahler1
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Homework Statement .
Let ##X## be a set and ##\mathcal A \subset \mathcal P(X)##. Prove that there is a topology ##σ(A)## on ##X## that satisfies
(i) every element of ##A## is open for ##σ(A)##
(ii) if ##\tau## is a topology on ##X## such that every element of ##\mathcal A## is open for ##\tau##, then ##σ(A) \subset \tau##
The attempt at a solution.
I suppose the idea is to construct the smallest possible topology which contains ##\mathcal A##
As the elements of ##\mathcal A## have to be open, then each ##U \in A## has to be in ##σ(A)##. Also, ##X,\emptyset## have to be in ##σ(A)##.
Since ##σ(A)## has to be closed under finite intersections and arbitrary unions, I'll have to add all sets that come from finite intersections of ##X## with elements of ##\mathcal A##.
So my idea was to define ##σ(A)=\{S \subset X : S=\bigcap_{i \in I} U_i, U_i \in \mathcal A, I \text{finite}\} \cup \{X\}## but when I've tried to show that ##σ(A)## was a topology I got stuck, so I don't know if that set works.
It is clear that ##X \in σ(A)##, and that if ##S_j \in σ(A)## for ##j \in J## with ##J## finite, then since ##S_j=\bigcap_{k=1}^{s_j} U_k##, ##\bigcap_{j \in J} S_j=\bigcap_{j \in J} (\bigcap_{k=1}^{s_j} U_k)##, which is clearly a finite intersection of elements in ##σ(A)##. The problem is with arbitrary unions, I couldn't show that arbitrary unions remain in ##σ(A)##. Where is my mistake?
Let ##X## be a set and ##\mathcal A \subset \mathcal P(X)##. Prove that there is a topology ##σ(A)## on ##X## that satisfies
(i) every element of ##A## is open for ##σ(A)##
(ii) if ##\tau## is a topology on ##X## such that every element of ##\mathcal A## is open for ##\tau##, then ##σ(A) \subset \tau##
The attempt at a solution.
I suppose the idea is to construct the smallest possible topology which contains ##\mathcal A##
As the elements of ##\mathcal A## have to be open, then each ##U \in A## has to be in ##σ(A)##. Also, ##X,\emptyset## have to be in ##σ(A)##.
Since ##σ(A)## has to be closed under finite intersections and arbitrary unions, I'll have to add all sets that come from finite intersections of ##X## with elements of ##\mathcal A##.
So my idea was to define ##σ(A)=\{S \subset X : S=\bigcap_{i \in I} U_i, U_i \in \mathcal A, I \text{finite}\} \cup \{X\}## but when I've tried to show that ##σ(A)## was a topology I got stuck, so I don't know if that set works.
It is clear that ##X \in σ(A)##, and that if ##S_j \in σ(A)## for ##j \in J## with ##J## finite, then since ##S_j=\bigcap_{k=1}^{s_j} U_k##, ##\bigcap_{j \in J} S_j=\bigcap_{j \in J} (\bigcap_{k=1}^{s_j} U_k)##, which is clearly a finite intersection of elements in ##σ(A)##. The problem is with arbitrary unions, I couldn't show that arbitrary unions remain in ##σ(A)##. Where is my mistake?