LEC ANA2 13

Connected spaces

Let $X$ be a topological space. Recall Def 144.1 and Def 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 a separation $A\bigsqcup B$ of $X$ to $-1$ and $+1$ respectively.□

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Corollary 329.2(Corollary).

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

Reviewed Thm 144.3, Thm 144.4, Thm 144.5.

Example 329.3(Example).

• $GL(2,\mathbb{R})$ is not collected; consider the determinant map in Thm 144.4.
• $O(2,\mathbb{R})=\{A\in GL(2,\mathbb{R}):A{A}^T=\text{id}\}$ is not connected.

Proposition 329.4(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. By Prp 1, the maps $f\circ i_y$ and $f\circ i_x$ are constant.

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

$$\begin{array}{r}f(x,y)=(f\circ i_y)(x)=(f\circ i_y)(x_0)=f(x_0,y)=(f\circ j_{x_0})(y)=(f\circ j_{x_0})(y_0)=f(x_0,y_0).\end{array}$$

Thus, $f$ is constant, and $X\times Y$ is connected.□

Proposition 329.5(Proposition).

Let $X$ be connected. $f:X\to \mathbb{R}$ be locally constant, that is, every $x\in X$ has an open neighborhood in $X$ on which $f$ is constant¹. Then, $f$ is constant.

Proof.

Fix $x_0\in X$. Define $V_{x_0}:=\{x\in X:f(x)=f(x_0)\}$. Clearly, $V_{x_0}$ and $V_{x_0}^c$ are open, that is, $V_{x_0}$ is clopen. Since $X$ is connected and $V_{x_0}$ is nonempty, $V_{x_0}=X$.□

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Saw Thm 187.3 as a corollary of Prp 5.

Footnotes

1. Note that this implies $f$ is continuous. ↩