another topology question

Question

zzr0ck3r · Answer

So another question for you, if you don't mind....

If we take some topology $X_{T_1}$, and  subbasis $S$ for this topology, and suppose we have a supspace $Y_{T_1}$ where $Y\subset X$. If we build a collection $S_2$ by taking an element in $S$ and its intersection with $Y$, for each element in $S$, then we can show that this is a subbasis for a topology on $Y$ buy showing that the union of $S_2$ is $Y$. 

Now $S_2$ is a subbasis for a topology on $Y$ but it is not neseceraly the case that this topology is the same as $Y_T$ (it is but nowhere above have I proved that). 

To show that it is the same topology, we can show that the collection of finite intersections of $S_2$ has the property that for  $U\in Y_T$ and each $x\in U$ there exists finite intersection of elements in $S_2$, call it $A=S_1\cap S_2\cap....\cap S_n$ such that $x\in A\subset U$. If show this, then we have shown that $S_2$ is a subbasis for $Y_T$?

note: my definition for a subbasis for $X$ is a collection that covers $x$. My definition for the topology it generates is the union of finite intersections of the subbasis. I have shown that a collection of sets with the property I mentioned above forms a basis for the topology in question. 

In general have I said anything that is not sound? 

@eliassaab

some.random.cool.kid · Answer

dag i hope i never reach this..

anonymous · Answer

i'll read ltr

zzr0ck3r · Answer

@Marki ?

zzr0ck3r · Answer

Hey @eliassaab I am to show that $\mathcal{T}_{X\times Y}$ is the smallest topology such that $\pi_X:X\times Y\rightarrow X, \pi_X(x)=x$, and $\pi_Y$ (defined similarly) are continuous.

What do they want me to show here?

anonymous · Answer

Read this https://www.proofwiki.org/wiki/Sub-Basis_for_Topological_Subspace