The most fundamental way to show two sets $A,B$ are equal is to show that each is a subset of the other.This is the definition of set equality.
However, we don't have to do this when the sets have some "structure".
The situations I can remember are below(all assumes $A\subset B$):
- They are finite sets and $|A|=|B|$
- They are finite dimensional vector spaces and $\dim A=\dim B$
- $B$ is a connected space, and $A$ is a non empty clopen set
Do you come up with other ways/situations? Please tell me as many as you can think of. Some kind of techniques are also welcome.
In topology it is often desirable to how that two topologies $T_1, T_2$ on a set are in fact the same topology. For example, we might exhibit bases $B_1, B_2$ for $T_1,T_2$ respectively, such that
(i) every $b\in B_1$ is a union of members of $B_2$ and every $b'\in B_2$ is a union of members of $B_1$
or (ii) whenever $c\in b\in B_1$ there exists $b'\in B_2$ such that $x\in b'\subset b,$ and whenever $x\in c\in B_3$ there exists $c'\in B_2$ such that $x\in c'\subset c.$
Or, if each $T_i$ is generated by a metric $d_i,$ we might show that any sequence is $d_1$-convergent iff it is $d_2$-convergent.