2
votes
3answers
456 views

Axiom of Choice and Right Inverse

I read an Theorem that states: Let $A$ and $B$ be non-empty sets, and let $f:A \to B$ be a function, then the function $f$ has a right inverse if and only if $f$ is surjective. The Theorem ...
2
votes
1answer
104 views

(Revisited$_2$) Injectivity Relies on The Existence of an Onto Function Mapping Back to Its Preimage

QUEST: For any sets $X$ and $Y$, there exists an injective function $f:X\rightarrow Y$ if and only if there exists a surjective function $g:Y\rightarrow X$. QUESTION$_1$: How do you people ...
-2
votes
1answer
90 views

Axiom of Choice: An Invocation Necessary for A Proof on Surjectivity [closed]

Prove that if $f\colon X\rightarrow Y$ is surjective, then there must exist a function $g\colon Y\rightarrow X$ such that $f\circ g=1_Y$, where $1_Y$ is the identity map on $Y$.
5
votes
2answers
833 views

Is there a Cantor-Schroder-Bernstein statement about surjective maps?

Let $A,B$ be two sets. The Cantor-Schroder-Bernstein states that if there is an injection $f\colon A\to B$ and an injection $g\colon B\to A$, then there exists a bijection $h\colon A\to B$. I was ...
9
votes
3answers
2k views

What is a basis for the vector space of continuous functions?

A natural vector space is the set of continuous functions on $\mathbb{R}$. Is there a nice basis for this vector space? Or is this one of those situations where we're guaranteed a basis by invoking ...
6
votes
3answers
540 views

What is the set-theoretic definition of a function?

I'm reading through Asaf Karagila's answer to the question What is the Axiom of Choice and Axiom of Determinacy, and while reading the explanation of Bertrand Russell's analogy ("The Axiom of Choice ...