# Confluent

Let $X$ and $Y$ be topological spaces. A function $f \colon X \rightarrow Y$ is called confluent if for every subcontinuum $Q$ of $Y$ and each component $C$ of $f^{-1}(Q)$ it follows that $f(C)=Q$.

# Properties

Theorem: If $f \colon X \rightarrow Y$ is onto and confluent, $B \subset Y$, and $A$ is the union of some components of $f^{-1}(B)$, then the restriction map $g = f \Big|_A \colon A \rightarrow f(A)$ is also confluent.

Proof:

Theorem: If $f \colon X \rightarrow Y$ is onto and confluent, and $A=f^{-1}(f(A))$, then the restriction map $g=f\Big|_A \colon A \rightarrow f(A)$ is confluent.

Proof:

Theorem: Let $f_1 \colon X \rightarrow Y$ and $f_2 \colon Y \rightarrow Z$. If $f \colon X \rightarrow Z$ is onto and confluent and $f=f_2 \circ f_1$, then $f_2 \colon Y \rightarrow Z$ is confluent.

Proof:

Theorem: Let $X$ be atroidic and $f \colon X \rightarrow Y$ confluent, then $Y$ is atroidic (or generally: contains no $n$-od).

Proof:

Theorem: Let $X$ and $Y$ be compact. Let $f \colon X \rightarrow Y$ be onto and an open map, then $f$ is confluent.

Proof:

Theorem: A confluent image of a $\lambda$-dendroid is a $\lambda$-dendroid.

Proof:

Corollary: Confluent map of a dendroid is a dendroid.

Proof:

Corollary: Confluent map of a dendrite is a dendrite.

Proof:

Corollary: Confluent map of a fan is a fan.

Proof:

Theorem: $Y$ is the confluent image of a Cantor fan if and only if $Y$ is Kelley.

Proof:

Theorem: Confluent map of an arc is an arc.

Proof:

Theorem: If $e$ is an endpoint of $Y$ and $f\ colon X \rightarrow Y$ is confluent, then there exists $a \in \mathrm{endpoints}(X)$ such that $f(a)=e$.

Proof:

Corollary: Let $L$ denote the Lelek fan. If $f \colon L \rightarrow Y$ is confluent, then $Y$ is homeomorphic to $L$.

Proof: