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Let $X_1$, $X_2$ be topological spaces. Let $f_1:Y\rightarrow X_1$ and $f_2:Y\rightarrow X_2$. I will construct a (fake) product $X_1\times X_2$.

Let $\pi_1$, $\pi_2$ be cartesian projections.

Let $f$ be the function defined by the equality $f(x,y)=(f_1(x),f_2(y))$.

Take $X_1\times X_2$ the discrete topology on $X_1\times X_2$.

For every object $Y$ and pair of morphisms $f_1: Y \to X_1$, $f_2: Y \to X_2$ there exists a unique function $f: Y \to X$ such that $\pi_1\circ f=f_1$ and $\pi_2\circ f=f_2$ ($f$ is morphism because every function from a discrete space is a morphism).

So discrete topology is a product.

Where is my error?

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For finite products the product topology on $\prod_{i=1}^n X_i$ where $X_i$ are discrete spaces should be the discrete topology. But for non finite products this is doesn't hold always – Dominic Michaelis Aug 31 '13 at 11:36
"there exists a unique function" <- That function ought to be continuous to have a product in the category of topological spaces with continuous mappings as morphisms. The function will in general not be continuous unless $X_1$ and $X_2$ are discrete. – Daniel Fischer Aug 31 '13 at 12:04
"Let f is the function defined by the equality f(x,y)=(f(x),f(y))"... Hmm, could you say it again? – Did Aug 31 '13 at 12:10
What is $X$? Is it $X_1\times X_2$? – Cameron Buie Aug 31 '13 at 12:24
@Daniel: Make it an answer? – Cameron Buie Aug 31 '13 at 12:27
up vote 3 down vote accepted

A product of two objects $X_1,\, X_2$ in a category is an object $X$ together with a pair of morphisms $\pi_i \colon X \to X_i$ such that for every object $Y$ and pair of morphisms $f_i \colon Y \to X_i$, there exists a unique morphism $f \colon Y \to X$ such that $f_i = \pi_i \circ f$.

In the category $\mathbf{Top}$, that means continuous maps.

Now, if you endow the cartesian product (the product in the category $\mathbf{Ens}$ of sets) $X_1\times X_2$ with the discrete topology, the function $f\colon x \mapsto (f_1(x), f_2(x))$ is the unique function with $\pi_i \circ f = f_i$, but it is in general not a morphism in $\mathbf{Top}$, i.e., it is in general not continuous (unless either one of the spaces is empty, or both spaces are discrete).

For example, if $X_1$ is not discrete, and $X_2$ not empty, we have the continuous maps $f_1 = \operatorname{id} \colon X_1 \to X_1$ and $f_2 \colon X_1 \to X_2;\; x \mapsto c$ for some $c \in X_2$. Then the map $x \mapsto (x,c)$ into the cartesian product $X$ with the discrete topology is not continuous.

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