# Existence of zero objects

Here are some exercises from Rotman's book (homological algebra).

$1$. Show that $\textbf{Top}$ has no zero object.

Let $X$ be a zero object in $\textbf{Top}$ then as X is initial and $\emptyset$ is a topological space then there is a unique continuous map $f: X \rightarrow \emptyset$. Therefore $X$ is the empty set, since $f$ is the empty function (which is continuous).

On the other hand, since $X$ is terminal choose a non-empty topological space $Z$. Thus there is a unique continuous map $f: Z \rightarrow \emptyset$ but there is no such function since $Z \neq \emptyset$.

$2$. Prove that the zero ring is not an initial object in the category of commutative rings.

Can we simply say: assume there is a ring homomorphism $f: 0 \rightarrow \mathbb{Z}$ where $0$ denotes, by abuse of notation, the zero ring. On one hand $f(0)=1$ because $f$ preserves the unity and $0=1$ in the zero ring. But on the other hand $f(0)=0$ so $0=1$ in $\mathbb{Z}$ which is absurd.

Is this OK? thanks for your time/help.

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In the second paragraph of 1., you mean "there is a unique continuous map $f: Z\rightarrow \emptyset$." – Alex Kruckman Dec 21 '12 at 3:44
@Alex Kruckman: right, thanks! – user10 Dec 21 '12 at 3:47