LEC ANA2 13

Connected spaces

Let $X$ be a topological space. Recall Definition 144.1 and Definition 144.2.

Proposition 329.1(Proposition).

$X$ is connected iff every continuous function $f:X\to \{\pm 1\}$, where $\{\pm 1\}$ has the discrete topology, is constant.

Proof.

Define$A=f^{-1}(1)$, $B=f^{-1}(-1)$. Clearly, $A\cap B=\varnothing$, $A\cup B=X$, and $A$ and $B$ are open. Thus, if $f$ is not constant, $X$ is not connected. If $X$ is not connected, you can define $f$ to map the separation of $X$ to $-1$ and $+1$.□

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Lemma 329.2(Lemma).

A set $J\subseteq \mathbb{R}$ is connected iff $J$ is an interval.

[!Proof]-
Suppose $J$ is not an interval. There exist $a,b\in J$ such that $a<c<b$ and $c\notin J$. $(-\infty ,c)\cap J$, $(c,\infty )\cap J$ are open in $J$ and form a separation of $J$.

Let $J$ be a finite interval $[a,b]$, and $J=(U\cap J)\cup (V\cap J)$ for open $V$ and $U$. Suppose $U\cap J\ne \varnothing$. Let $c\in U\cap J$ such that $a<c<b$. There exists $ϵ>0$ such that $(c-ϵ,c+ϵ)\cap J\subseteq U\cap J$, so $[c,c+ϵ)\subseteq U\cap J$. Let $x_0=\sup \{x:[c,x)\subseteq U\cap J\}$. $x_0\ne b$ leads to a contradiction; we must have $x_0=b$ and $b\in U\cap J$.

[!Example]

• $GL(2,\mathbb{R})$ is not collected; Trivial by Theorem 144.4.
• $O(2,\mathbb{R})=\{A\in GL(2,\mathbb{R}):A{A}^T=\text{id}\}$ is not connected.

[!Proposition]
Let $A,B\subseteq X$ be connected. If $A\cap B\ne \varnothing$, then $A\cup B$ is connected.

[!Proof]-
Use Proposition 1.

[!Proposition]
If $X$ and $Y$ are connected, $X\times Y$ is connected.

[!Proof]-

Pick some continuous $f:X\times Y\to \{\pm 1\}$. For every $y\in Y$, define

$$i_y:X\to X\times Y,x\mapsto (x,y)$$

and for every $x\in X$, define

$$j_x:Y\to X\times Y,y\mapsto (x,y).$$

These are continuous families. Thus, the maps $f\circ i_y$ and $f\circ i_x$ are constant.

Fix $(x_0,y_0)\in X\times Y$. Then

[!Proposition]
Let $X$ be connected. $f:X\to \mathbb{R}$ be locally constant(implies continuous), that is, every $x\in X$ has an open neighborhood in $X$ on which $f$ is constant. Then, $f$ is constant.

[!Proof]-
Show that $V_{x_0}:=\{x\in X:f(x)=f(x_0)\}$ is clopen.

[!Corollary]
Let $U\subseteq {\mathbb{R}}^n$ be open and connected. $f:U\to R$ be differentiable. If $\mathbf{D}f(p)=0$ for all $p\in U$, then $f$ is continuous.