Differences

This shows you the differences between two versions of the page.

--- math104-s22:notes:lecture_15 [2022/03/08 07:03]
pzhou
+++ math104-s22:notes:lecture_15 [2026/02/21 14:41] (current)
@@ Line 31: / Line 31: @@
 . We know that $[0,1]$ is sequentially compact (by Heine-Borel theorem), can you show that $[0,1]^2$ is sequentially compact? (Hint: given a sequence $(p_n)$ in $[0,1]^2$, first show that you can pass to subsequence to make the sequence of the first coordinate converge, then ...)
-===== Connectedness =====
-Have you wondered, what subset of a metric space is both open and closed?
-We say a metric space $X$ is connected, if $X$ cannot be written as disjoint union of two non-emtpy open subset. In other word, the only subsets in $X$ that is both open and closed are $X$ and $\emptyset$.
-For example, $\Q$ is not connected, since $(-\infty, \sqrt{5})$ is both open and closed in $Q$. (why?)
-For example, $X=\{1,2,3\}$ (with induced metric from $\R$) is not connected, since $\{1\}$ is both open and closed in $X$. (Discussion: Equip $X$ with the induced metric, can you show that $\{1\}$ is both open and closed? Equip $X$ with the induced topology, can you show that $\{1\}$ is both open and closed? )
-Theorem: a subset $E \In \R$ is connected, if and only if, for any $x,y \in E$, we have $[x,y] \In E$. \\
-Proof: next time.

Lecture Notes