One las thing If $A\ U$ and $A$ is non empty then $G= \{B\subset U: A\subset B\}\cup\{\varnothing\}$ is clearly a topology. But the exercises says what is the smallest base for the topology. mmm I think that are just the proper subset containing A ...
@MikeMiller the problem said Prove that if X and Y are compact Hausdorff spaces and if f is a function from X to Y whose graph is closed in X × Y , then f is continuous.
I think a found other mistake in the notes. This says: Let $\varnothing \subset A_1 \subset A_2 \.ldots A $ where $A$ is the union and the inclusions are proper. SO $T=\{A_ns, A, \varnothing and X\}$ is clearly a topology.
BUt in one point this says Let $Y\subset X$ then Y is compact iff on of the following statements holds. 1 $Y\subset A_N$ for some N of exists $x\in Y \setminus A$
So we have $X \times Y \cap X \times C$. This is closed? And in a compact space, so it's compact. Moreover, the projection of this set onto $X$ is continuous for the product topology, and continuous maps take compact sets to compact sets, so $f^{-1}(C)$ is compact. Since it's compact and Hausdorff, it's closed. So the preimage of closed is closed, and our function is continuous.
So yeah, $X \times Y$ just means the whole space. You should have just called the graph $G$ or something; $(X,Y)$ doesn't make much sense. Other than notation, you're golden.
the return does not work for $Y=2\mathbb{N}\cup \{0\}$ which can be cover by the $A_n$s and the $X$ this satisfies the second condition but does not have a finite subcover.
I think a found other mistake in the notes. This says: Let $\varnothing \subset A_1 \subset A_2 \.ldots A $ where $A$ is the union and the inclusions are proper. SO $T=\{A_ns, A, \varnothing and X\}$ is clearly a topology. But in the notes this says Let $Y\subset X$ then Y is compact iff on of the following statements holds. 1 $Y\subset A_N$ for some N of exists $x\in Y \setminus A$. One part is really trivial, but the return does not work
for $Y=2\mathbb{N}\cup \{0\}$ which can be cover by the $A_n$s and the $X$ this satisfies the second condition but does not have a finite subcover.
Assuming $X=\mathbb{N}$ andlet the sets $\varnothing \subset \{o\}\subset \{0,2\}\ldots$ the even numbers
@Anthony Generally, the projections from a product to the factors are not closed. And if $X$ is not quasicompact, then you only know that $X\times C$ is closed in $X\times Y$, and hence that $X\times C \cap \Gamma(f)$ is closed, not that it is quasicompact. So the quasicompactness of $X$ and that every quasicompact subset of $X$ is closed, are also important.
