# Maximal connected component.

Let $f: X \to Y$ an application between two topological spaces $\mathbb{X}$ and $\mathbb{Y}$ both Hausdorff. We know that if $Y \in\mathbb{Y}$ is connected and $f$ is continuous then $f^{-1}(Y)$ is connected.

But if $A$ is not connected or if $f$ is not continuous then $f^{-1}(Y)$ is not necessarily connected.

And in this case it is customary to work with the so-called maximal connected component.

Question 1: What is maximal connected component?

Add after Azarel Answer the felowing question.

Question 2: What is the maximal connected component of $f^{-1}(y)$ whit $y\in\mathbb{Y}? - It is not true that$f^{-1}(Y)$is connected if$Y$is. Take the constant map$[0,1]\cup [2,3]$to$1$. The point$1$is connected, but$f^{-1}(1)$is not. – Joe Johnson 126 Mar 9 '12 at 17:21 You are wrong the image of something connected is connected and not the preimage. Map two points to one point. – plusepsilon.de Mar 9 '12 at 17:22 ## 2 Answers You have a function$f:\Bbb X\to \Bbb Y$and a point$y\in\Bbb Y$. Let$A=f^{-1}[\{y\}]$. A maximal connected component of$A$is a connected subset$C$of$A$with the property that if$C\subseteq S\subseteq A$, and$S$is connected, then$C=S$. In other words, no connected subset of$A$properly contains$C$. You can find the maximal connected components of$A$in the same way that you find the maximal connected components of a whole space, as given in azarel’s answer. For each$x\in A$let $$\mathscr{C}_x=\{C\subseteq A:x\in C\text{ and }C\text{ is connected}\}\;,$$ and let $$C_x=\bigcup\mathscr{C}_x=\bigcup\{C:C\in\mathscr{C}_x\}=\bigcup_{C\in\mathscr{C}_x}C\;.$$ Then each$C_x$is a maximal connected component of$A$, and every maximal connected component of$A$is obtained in this way. - Let's$A_1\cap A_2=\emptyset$,$A_2\cap A_3=\emptyset$and$A_3\cap A_1=\emptyset $. Let's$A=A_1\cup A_2 \cup A_3$and$1_A : \mathbb{X}\to\mathbb{Y}$the caracterisc function of$A$. What is the maximal connected component of$1_A^{-1}(1)$? – MathOverview Mar 9 '12 at 18:14 @Elias: They are the connected components of$A_1$,$A_2$, and$A_3$. If$A_1$,$A_2$, and$A_3$are connected, then they are the connected components of that inverse image. – Brian M. Scott Mar 9 '12 at 18:16 The maximal connected component need not be a set connected? This is what you mean? – MathOverview Mar 9 '12 at 18:23 @Elias: No, a maximal connected component is by definition connected. – Brian M. Scott Mar 9 '12 at 18:29 For every$x\in X$consider the set$\mathcal C_x=\{C: x\in C, \ \text{and }\ C\ \text{is connected}\}$. It is easy to see that$\bigcup \mathcal C_x$is connected. Moroever, it is maximal in the sense that it is the biggest connected subset containing$x$. The sets$\bigcup\mathcal C_x\ (x\in X)$are the maximal connected components of the space. - Your notation is a bit confusing to me.$\mathcal{C}_x$is a set of subsets. The union of all subsets in$C_x$is written$\bigcup_{U \in \mathcal{C}_x} U$or somesuch. And your last sentence should be something like "The sets$\bigcup_{U \in \mathcal{C}_x} U (x \in X)$are the maximal connected components of the space." The maximal component containing$x$is a subset of$X$rather than a set of subsets. Apologies if I'm just showing off my ignorance about notation (or topology). – Adam Saltz Mar 9 '12 at 17:29 I don't understend you answer. For exemple, what is the maximal connected component of$f^{-1}(y)$whit$y\in\mathbb{Y}$? – MathOverview Mar 9 '12 at 17:33 @IAmBrianDawkins:$\bigcup\mathscr{C}_x$is a perfectly standard notation for the union of the members of the collection$\mathscr{C}_x$. The$\mathscr{C}_x$in the last sentence should of course be$\bigcup\mathscr{C}_x\$. – Brian M. Scott Mar 9 '12 at 17:46
@BrianM.Scott, thanks very much. I wasn't familiar with that. – Adam Saltz Mar 9 '12 at 17:51