Question concerning possible typo on HW (Topology)

[BSMSMSTMSPHD]

I'm trying to prove the following Theorem.

Suppose T1 and T2 are topologies for X. The following are equivalent:

1. T1 is a subset of T2;

2. if F is closed in (X, T1), then F is closed in (X, T2);

3. if p is a limit point of A in (X, T2), then p is a limit point of A in (X, T1).

So far, I've shown 1 implies 2. However, I'm curious about the reversal of T1 and T2 in statements 2 and 3. Is there a typo? I'm tying to show 1 implies 3, but I'm having no luck. I'll try reversing statement 3 and seeing if that works. If anyone thinks that there is a typo, please let me know. Thanks!

 

[matt grime]

Topologies are about open things, usually, limit points are abut closed things. Taking complements, i.e. switching from open to closed changes the order of containment. This is called contravariance, and is very important. N.B. I've not checked you particular case, but am just explaining a general principle.