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Thread: Intersection of 2 topologies is a topology proof

  1. #1
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    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)$\displaystyle T_i \cap T_j = ( T_{11} \cap T_{12}) \cap (T_{21} \cap T_{22}) $ where $\displaystyle T_{11}, T_{12} \in T_i, T_{21}, T_{22} \in T_j$
    $\displaystyle = ( T_{11} \cap T_{21}) \cap (T_{12} \cap T_{22}) $
    $\displaystyle = \in T_i, \in T_j$

    I cannot see how to get (T3) to work
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  2. #2
    Super Member girdav's Avatar
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    Re: Intersection of 2 topologies is a topology proof

    Take $\displaystyle O_i$, $\displaystyle i\in I$, sets of $\displaystyle T_1\cap T_2$. Since for all $\displaystyle i$, $\displaystyle O_i\in T_1$ then $\displaystyle \bigcup_{i\in I}O_i\in T_1$ and now show that $\displaystyle \bigcup_{i\in I}O_i\in T_2$.
    If you take the set $\displaystyle X=\{a,b,c\}$ and $\displaystyle T_1=\{\emptyset,\{a\},X\},$ $\displaystyle T_2=\{\emptyset,\{b\},X\}$, what about $\displaystyle T_1\cup T_2$?
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  3. #3
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    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?
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  4. #4
    Super Member girdav's Avatar
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    Re: Intersection of 2 topologies is a topology proof

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