# Thread: Intersection of 2 topologies is a topology proof

1. ## Intersection of 2 topologies is a topology proof

Show that the intersection of two topologies on the same set X is also a topology on X, but that their union may or may not be a topology.

I think this is way easier than i think. I have tied my mind in knots.

You need to check (T1-T3)

(T1) Obv from def of topology
(T2) $T_i \cap T_j = ( T_{11} \cap T_{12}) \cap (T_{21} \cap T_{22})$ where $T_{11}, T_{12} \in T_i, T_{21}, T_{22} \in T_j$
$= ( T_{11} \cap T_{21}) \cap (T_{12} \cap T_{22})$
$= \in T_i, \in T_j$

I cannot see how to get (T3) to work

2. ## Re: Intersection of 2 topologies is a topology proof

Take $O_i$, $i\in I$, sets of $T_1\cap T_2$. Since for all $i$, $O_i\in T_1$ then $\bigcup_{i\in I}O_i\in T_1$ and now show that $\bigcup_{i\in I}O_i\in T_2$.
If you take the set $X=\{a,b,c\}$ and $T_1=\{\emptyset,\{a\},X\},$ $T_2=\{\emptyset,\{b\},X\}$, what about $T_1\cup T_2$?

3. ## Re: Intersection of 2 topologies is a topology proof

Sorry i am confused as to your reply.

What are you trying to say in your first line?

4. ## Re: Intersection of 2 topologies is a topology proof

I only check that each union of elements of $T_1\cap T_2$ is in $T_1\cap T_2$.

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# prove that intersection of two topologies is also topology, but union of two topologies is not necessary a topology

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