Subspace/induced/relative Topology Definition

[filter54321]

I'm having trouble understanding the definition of a Subspace/Induced/Relative Topology. The definitions I'm finding either don't define symbols well (at all).

If I understand correctly the definition is:

Given:
-topological space (A,$$\tau$$)
-$$\tau$$={0,A,u₁,u₂,...u_n}
-subset B$$\subset$$A

The subspace topology on B will be the intersection of B and every part of the topology of A

OR

$$\tau$$_B={0,B,B$$\bigcap$$u₁,B$$\bigcap$$u₂,...B$$\bigcap$$u_n}


...I apologize in advance for my LATEX work.

 

[Hurkyl]

That looks correct idea for the subspace topology... but there is a serious problem in your understanding of topology: you seem to be assuming there are only finitely many open sets.

 

[filter54321]

The u's are finite for illustrative purposes. I wanted to avoid using the form given by Wikipedia because it has insufficient textual explanation.

They have this definition, but don't specify exactly what a U is:

$$\tau$$_B={B$$\bigcap$$U|U$$\in$$$$\tau$$}

It seems that me that you need to say how U fits into the first topology.

 

[Hurkyl]

They have this definition, but don't specify exactly what a U is:

Yes they do: they say a U is an element of $\tau$.

This is standard set-builder syntax for replacement: on the right of | you introduce a variable and its domain, and on the left of the | you have a function of that variable indicating what should go into the set you're building.