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Show that if $(X_i)_{i \in \mathcal I}$ where $X_i$ is a topological space for every $i \in \mathcal I$, then $X_i$ is connected for every $i$ if and only if $\prod_{i \in \mathcal I} X_i$ is connected.

I could do one implication: If I define the projection map $\Pi_i:\prod_{i \in \mathcal I} X_i \to X_i$, then since $\Pi_i$ is continuous, it follows $X_i$ is connected for every $i \in \mathcal I$.

I need help to prove that if each $X_i$ is connected, then $\prod_{i \in \mathcal I} X_i$ is connected. Thanks in advance.

user156441
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2 Answers2

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First prove that any finite product of connected spaces is connected. This can be done by induction on the number of spaces, and for two connected spaces $X$ and $Y$ we see that $X \times Y$ is connected, by observing that $X \times Y = (\cup_{x \in X} (\{x\} \times Y)) \cup X \times \{y_0\}$, where $y_0 \in Y$ is some fixed point. This is connected because every set $\{x\} \times Y$ is homeomorphic to $Y$ (hence connected) and each of them intersects $X \times \{y_0\}$ (in $(x,y_0)$), which is also connected, as it is homeomorphic to $X$. So the union is connected by standard theorems on unions of connected sets. Now finish the induction. So for every finite set $I$, $\prod_{i \in I} X_i$ is connected.

Now if $I$ is infinite, fix points $p_i \in X_i$ for each $i$, and define $$Y = \{ (x_i) \in \prod_i X_i: \{i \in I: x_i \neq p_i \} \text{ is finite }\}$$

Now for each fixed finite subset $F \subset I$, define $Y_F = \{ (x_i) \in \prod_i X_i: \forall i \notin F: x_i = p_i \}$. By the obvious homeomorphism, $Y_F$ is homeomorphic to $\prod_{i \in F} X_i$, which is connected by the first paragraph. So all $Y_F$ are connected, all contain the point $(p_i)_{i \in I}$ of $\prod_{i \in I} X_i$, and their union (over all finite subsets $F$ of $I$) equals $Y$. So again by standard theorems on the union of connected subsets of a space, $Y$ is a connected subspace of $\prod_{i \in I} X_i$.

Finally note that $Y$ is dense in $\prod_{i \in I} X_i$, because every basic open subset $O$ of the product depends on a finite subset of $I$, in the sense that $O = \prod_{i \in I} U_i$ where all $U_i \subset X_i$ are non-empty open and there is some finite subset $F \subset I$ such that $U_i = X_i$ for all $i \notin F$. Pick $q_i \in U_i$ for $i \in F$ and set $q_i = p_i $ for $i \notin F$. The $(q_i)_{i \in I}$ is in $O \cap Y_F \subset O \cap Y$, so every (basic) open subset of $\prod_{i \in I} X_i$ intersects $Y$.

Now use that the closure of a connected set is connected to conclude that $\prod_{i \in I} X_i$ is connected, also for infinite $I$.

Henno Brandsma
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  • Why the union of $Y_F$ is equal to $Y$ ? Could you explain please ? –  Feb 25 '18 at 06:15
  • @bella It's by definition: Let $y \in Y$. Then define $F_y = {i \in I: y_i \neq p_i }$ which is finite by definition of $y$ being in $Y$. Then again by definition, $y \in Y_{F_i}$. So $y \in \bigcup {Y_F: F \subseteq I \text { finite }}$. $Y$ is sometimes called the $\sigma$-product w.r.t. $(p_i)_i$, BTW. – Henno Brandsma Feb 25 '18 at 07:17
  • Henno my brain is going to explote –  Feb 25 '18 at 07:36
  • What is $Y_{F_i}$? –  Feb 25 '18 at 07:37
  • @bella I meant $Y_F$ for $F = F_y$, not $F_i$. so $Y_{F_y}$, really. typo. – Henno Brandsma Feb 25 '18 at 08:17
  • @HennoBrandsma Unfortunately I don't understand why $Y_F$ is homeomorphic to $\prod_{i\in F}X_i$, so could you explain to me this, please? – Antonio Maria Di Mauro Apr 10 '20 at 15:19
  • @HennoBrandsma I try to define a function $\phi:Y_F\rightarrow\prod_{i\in F}X_i$ through the condiction $$ \phi(x):=x(i) $$ for each $i\in F$. So obviously for each $i\in F$ it follows that $\pi'i\circ \phi=\pi_i$, where $\pi'_i$ and $\pi_i$ are respectively a projection in $\prod{i\in F}X_i$ and in $\prod_{i\in I}X_i$, and so we can conclude that $\phi$ is continuous. – Antonio Maria Di Mauro Apr 10 '20 at 15:54
  • Then we define $\psi:\prod_{i\in F}X_i\rightarrow Y_F$ through the condiction $$ \psi(x):=\begin{cases}x(j),\quad\text{if}\quad j\in F\p_j,\quad\text{otherwise}\end{cases} $$ and so we observe that $\phi\circ\psi=\text{id}{\prod{i\in F}X_i}$ and $\psi\circ\phi=\text{id}_{Y_F}$ and so $\phi$ is a bjection and $\psi=\phi^{-1}$. – Antonio Maria Di Mauro Apr 10 '20 at 15:54
  • Then we define $\Phi:\prod_{i\in F}X_i\rightarrow\prod_{i\in I}X_i$ through the condiction $$ \Psi(x):=\psi(x) $$ and so we observe that $$ \pi_j\circ\Psi=\begin{cases}\pi_j,\quad\text{if}\quad j\in F\ p_j,\quad\text{otherwise}\end{cases} $$ and so we conclude that $\Psi$ is continuous and so too $\psi$. Well we can conclude that $\phi$ is an homeomorphism. Is it right? – Antonio Maria Di Mauro Apr 10 '20 at 15:54
  • @AntonioMariaDiMauro That $\Psi$ is correct and has the projection $\pi_J$ as its continuous inverse. That's the obvious homeomorphism I was referring to. – Henno Brandsma Apr 10 '20 at 16:07
  • @HennoBrandsma Okay. Furthermore when you claim that ${x}\times Y$ and $X\times{y_0}$ are homeomorphic to $Y$ and $X$, do you naturally suggest that the topologies on ${x}$ and ${y_0}$ are the subspace topology induced by $X$ and $Y$, right? – Antonio Maria Di Mauro Apr 10 '20 at 20:37
  • @HennoBrandsma Rereading your answer I had one doubt. We know that if $\mathcal{A}={A_i:i\in I}$ is a collection of connected spaces ad if $\bigcap\mathcal{A}\neq\varnothing$ then $\bigcup\mathcal{A}$ is connected. So we know that $X\times{y_0}$ and ${x}\times Y$ are connected but $\bigcap_{x\in X}\Big(({x}\times Y)\cap(X\times{y_0}\Big)=\varnothing$. So how can I conclude that $\bigcup_{x\in X}\Big({x}\times X\Big)\cup\Big(Y\times{y_0}\Big)$ is connected? Could you explain this to me, please? – Antonio Maria Di Mauro Apr 27 '20 at 10:37
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    @AntonioMariaDiMauro I use the more general theorem (what we called the "glue theorem" in class a long time ago): if all $A_i$ are connected and $B$ is connected and $A_i \cap B \neq \emptyset$ for all $i$ ($B$ is the "glue" that glues the $A_i$ together) then $B \cup \bigcup_{i \in I} A_i$ is connected. (The common intersection point $B={x_0}$ can be used the case where $x_0 \in \bigcap_i A_i$.) This more general fact is easy to prove and should be standard in all text books, not just my old college notes. That's what I meant by standard theorems on unions of connected subspaces. – Henno Brandsma Apr 27 '20 at 10:56
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    @AntonioMariaDiMauro Proof: if $f: A:= B \cup \bigcup_i A_i \to {0,1}$ is continuous, then $f\restriction B$ is constant, say with value $i_0$. Then for any $i$, $f\restriction A_i$ is constant too by connectedness of $A_i$ and its value must be $i_0$ from its common point with $B$. So $f \equiv i_0$ on the union $A$ and QED. – Henno Brandsma Apr 27 '20 at 11:21
  • All clear, thanks!!! – Antonio Maria Di Mauro Apr 27 '20 at 13:28
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Hint: Suppose that $F: \prod X_i \to \{0,1\}$ is continuous. For some choice of $x_j \in X_j$ for all $j \neq i$, we can define a continuous function from $X_i \to \{0,1\}$ by $$ f(y) = F\left(\prod_{j \in I} y_j\right) $$ Where $y_j = x_j$ when $j \neq i$ and $y_i = y$.

Now, if $X_i$ is connected, then $F$ must be constant on every set of the form $f(X_i)$ with $f,F$ as described above.

Ben Grossmann
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