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Is my proof that the intersection of any family of topologies on a set $X$ is a topology on $X$ correct?

Proof. We are required to show that the intersection satisfies the topology axioms. Let $\tau$ be an arbitrary intersection of topologies on $X$.

  1. $\emptyset$ and $X$ are in every topology so they are in $\tau$
  2. Let $U=\bigcup_{i\in I} A_i$ be an arbitrary union of elements of the intersection $\tau$. $U$ is open in every topology (because $A_i$ is in $\tau$ for all $i\in I$) so it's open in $\tau$
  3. Let $V=A_1\cap\dots\cap A_n$ be a finite intersection of elements of the intersection. $V$ is open in every topology (because $A_i$ is in $\tau$ for all $i\in[1,n]$) so it's open in $\tau$

Moreover, $\tau\subseteq\tau_j$ for every topology $\tau_j$ in the intersected family so it's coarser than all of them.

Martin Sleziak
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mrk
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    I would include a bit more detail for 2 and 3. For instance "Each $A_i$ is open in every topology, so $\bigcup A_i$ is open in every topology; hence $\bigcup A_i$ is open in $\tau$." – Nick Peterson Jul 16 '13 at 13:07
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    Usually, the definition of topology postulates only the case $n=2$ in 3. Sometimes this simplifies arguments slightly. – Hagen von Eitzen Jul 16 '13 at 13:44

2 Answers2

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The proof seems just fine.${}{}$

Asaf Karagila
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Here is how I understand it.

Condition 1) is trivially satisfied since X and empty set are in every topology they will be in the intersection as well.

Condition 2) and 3) Here we need to prove that arbitrary unions and finite intersections of the elements in the intersection will be in the intersection. We explain this using intersection of two topologies over $X$. Let $T_1$ and $T_2$ be two topologies on $X$. Let set $A$ and set $B$ are in the intersection of $T_1$ and $T_2$. Since $A$ and $B$ are in the intersection of $T_1$ and $T_2$ it implies that $A$ and $B$ are both in $T_1$ and $T_2$ considered separately. Since $T_1$ and $T_2$ are topologies and $A$ & $B$ are member of both of these topologies, both $T1$ and $T2$ will contain union (and intersection) of $A$ & $B$ (by definition of being topologies) . Thus the union (and intersection) of $A$ & $B$ will be present in the intersection of the topologies $T_1$ and $T_2$. Hence we proved both condition 2 and condition 3 of being a topology.

This argument can be extended to the intersection of all the families of topologies on X.

Ananyapam De
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Shrey
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