## 倡议

（参考我的帖子：经验一二三给学数学的你中国的学生差在哪哦大学数学书等等

Tao在数学研究中的经验以及给年轻人的建议

Exercise 14 of the Notes 8 of the course: Real Analysis:Let X be a topological space. Show that X is compact if and only if every net has a convergent subnet. (Hint: equate both properties of X with the finite intersection property, and review the proof of Theorem 1). Similarly, show that a subset E of X is relatively compact if and only if every net in E has a subnet that converges in X. (Note that as not every compact space is sequentially compact, this exercise shows that we cannot enforce injectivity of $\phi$ in the definition of a subnet.

ProofWe only prove the first conclusion.

$\Rightarrow$“: Suppose for contradiction that the net $\{x_\alpha\}_{\alpha\in A}$ is without convergent subnet. Then an easy implication shows for any $x_0\in X$, there exists an open neighborhood $V_{x_0}$ of $x_0$ such that some $\alpha\in A$ satisfies that for all $\beta \geq \alpha$, $x_\beta\notin V_{x_0}$. All the $V_x$ form a open cover of the space X, and thus give rise to a finite number of $V_x$ which is a finite cover. Accordingly there exists a $\alpha_x$ for each x. Then since we can choose a $\alpha^*$ larger than each $\alpha_x$, the contradiction follows since there is no place to live for all the elements larger than $\alpha^*$.

$\Leftarrow$“: Suppose for contradiction that $(F_\alpha)_{\alpha \in A}$ is a collection of closed subsets of X such that any finite subcollection of sets has non-empty intersection, and the intersection of the entire collection is empty. Then choose from all the finite subcollection of sets $(F_\alpha)_{\alpha \in A}$ an element $x_{\alpha_1,\cdots \alpha_k}$. Define $(\alpha_1,\cdots,\alpha_k)\leq (\alpha_1',\cdots,\alpha_l')$ iff $\cap_{i=1}^kF_{\alpha_i}\subseteq \cap_{j=1}^lF_{\alpha_j'}$ and this forms a net. Observe that this net has no convergent sebnet, otherwise there would be some x in all the $F_\alpha$, a contradiction.

Bourbaki学派的最初组织人就是一群年轻人，提起他们这群人我是什么意思，我想读者比我更明白，呵。

Explore posts in the same categories: Maths

1. percy li Says:

i am a Chinese exchange student in Washington.d.c. i go to terry’s blog a lot but these math was too abstract for me. I once mentioned one of the typo in his book “sloving mathematical problems” and he really replied me. he was a great teacher. i would love to join the group if i were 5 years older…

2. liuxiaochuan Says:

Dear percy li:
The way that Professor Tao helps the Maths-learners around all over the world, in my opinion, is something just like Euler centuries ago. But the chinese students are seemingly less concerned.

As to your questions, I suggest you just try learning the posts which you are most interested. From my own experience, though it is quite difficult form the begining, you can learn much more during the process. Also, your will be more confident after some time. Anyway, you can just e-mail me at any time.

3. percy li Says: