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--- math104-s22:notes:lecture_15 [2022/03/07 23:03]
pzhou
+++ math104-s22:notes:lecture_15 [2022/03/09 22:37] (current)
pzhou [Connectedness]
@@ 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