Point Set Topology: Non-Trivial Facts Beyond Uryshon's Lemma

[facenian]

Hello
I am curious about this. Uryshon' s lemma is also known as "the first non-trivial fact of point set topology", what are the others non-trivial facts of point set topology?
I suppose Tychonoff' s theorem is another one.

 

[micromass]

I would say things like Tychonoffs theorem, Tietze's extension theorem, Cech-Stone compactifications,...

 

[micromass]

Not to forget space-filling curves and the theorem that says that every compact metric space is the image of the Cantor set. Also, the Siefert-Van Kampen theorem, but that's algebraic topology...

 

[Landau]

Perhaps the existence of a partition of unity for certain spaces.

 

[micromass]

Yeah, that to. And also Stones theorem that metric spaces are paracompact...

 

[facenian]

Thank you guys