# Product of connected sets is connected

Mathematics Asked by pipita on January 3, 2022

I know that this question had already been asked here but there is a problem … all the proofs that I’ve seen used homeomorphisms and continuous functions. Well my teacher didn’t teach me what a homeomorphism is so I cannot use the concept.

The question is:

Show that the product of connected sets $A$ in $X$ and $B$ in $Y$ is connected for the product topology.
Hints:
a) If $b$ belongs to $Y$, the product $Atimes{b}$ is connected
b) Using a), the product $Atimes B$ is connected

Well, my try:
I know that $Atimes B$ is connected if the only partition of the set is the trivial.
But I don’t know how to relate this with the hints … Can somebody give me more hints, please?

Since the 'product of subspaces = subspace of products' (see this), the question boils down to showing that the product of two connected spaces is connected. The answer to that question, using the OP's hints, can be found here.

Answered by CopyPasteIt on January 3, 2022

Yes, you just need the definition of homeomorphisms to prove the statement. I assume you know this fact 1: Let $S,T$, $S subset T$ two subset of a topological space $X$, assume that $S$ is connected and $T$ is not connected. If $(U,V)$ is a separation of $T$ then $S subset U$ or $S in U$. Here is a proof of the following fact2: Let X and Y two topological spaces, then $X times Y$ is connected $Leftrightarrow X,Y$ are connected. $(Leftarrow)$ suppose that $X,Y$ are connected and $(U,V)$ is a couple of open nonempty sets which unconnect $X times Y$ which means $X times Y= U bigcup V$. Now let $(x_0,y_0) in U$. The set ${x_0} times Y$ is connected because it is homeomorphic to $Y$(it is very easy to show!). ${x_0}times Y$ contains $(x_0,y_0)$ then for fact 1 ${x_0} times Y subset U$. On the other hand $forall y in Y$, the set $X times y$ is connected because is homeomorphic to X and contains $(x_0,y) in U$. Again for fact 1 we can deduct that $X times y subset U$. Then $X times Y= bigcup_{y in Y} X times y subset U$ $Rightarrow$ $X times Y=U$ and $V$ is empty, which is absurd because we supposed that $V$ is nonempty. This contradiction shows that $X times Y$ can't be unconnected then is connected. $(Rightarrow)$ Assume that $X times Y$ is connected, define the canonical projection $pi: X times Y rightarrow X$, $pi$ is surjective and a continuous map, and for the principal theorem of connection you easily conclude that $X$ is connected. The same reasoning shows that also $Y$ is connected. $Box$ The principal theorem of connection says that if $X,Y$ are a topological spaces with X connected and $f: X rightarrow Y$ is a continuous map, then $f(X)$ is connected.

Answered by Salvatore on January 3, 2022

## Related Questions

### (proposed) elegant solution to IMO 2003 P1

0  Asked on January 18, 2021 by mnishaurya

### Reverse order of polynomial coefficients of type $left(r-xright)^n$

1  Asked on January 18, 2021 by thinkingeye

### Should one include already cemented proofs of related principles in one’s paper?

0  Asked on January 18, 2021 by a-kvle

### The Cauchy-Crofton formula on a plane

1  Asked on January 18, 2021 by jalede-jale-uff-ne-jale

### Solving a limit for capacity of a transmission system

3  Asked on January 18, 2021 by jeongbyulji

### Galois group Abstract algebra

1  Asked on January 18, 2021 by user462999

### Does there always exists coefficients $c,dinmathbb{R}$ s.t. $ax^3+bx^2+cx+d$ has three different real roots?

2  Asked on January 17, 2021 by w2s

### If there is a “worldly ordinal,” then must there be a worldly cardinal?

2  Asked on January 17, 2021 by jesse-elliott

### Cover number and matching number in hypergraphs.

1  Asked on January 17, 2021 by josh-ng

### $displaystyleint_C (e^x+cos(x)+2y),dx+(2x-frac{y^2}{3}),dy$ in an ellipse

1  Asked on January 17, 2021 by fabrizio-gambeln

### Show that $h_n(x)=x^{1+frac{1}{2n-1}}$ converges uniformly on $[-1, 1]$.

1  Asked on January 17, 2021 by wiza

### What does the functional monotone class theorem say and how does it relate to the other monotone class theorem?

0  Asked on January 17, 2021 by alan-simonin

### Symmetric matrix and Hermitian matrix, unitarily diagonalizable

2  Asked on January 17, 2021

### direct conversion from az/el to ecliptic coordinates

0  Asked on January 16, 2021 by klapaucius-klapaucius

### A proof problem in mathematical statistics

2  Asked on January 16, 2021

### How to find the z component of the parameterization of an ellipse that is the intersection of a vertical cylinder and a plane

1  Asked on January 16, 2021 by nono4271

### using Lagrange multipliers to determine shortest distance between a point and straight line

2  Asked on January 16, 2021 by am_11235

### Proof of Lemma 5.1.5.3 in Jacob Lurie’s HTT.

1  Asked on January 16, 2021 by robin-carlier