1
$\begingroup$

So I basically have the following. Let $OP(-):\textbf{Top}\rightarrow \textbf{Cat}$, be given by $X\mapsto OP(X)$ where $OP(X)$ is defined as the category whose objects are open subsets $U$ of $X$, and whose morphisms are defined as $Mor(U,V)=\{incl_{VU}\}$ if $U\subset V$ and $\emptyset$ otherwise.

The problem is asking to show that $OP(-)$ is a contravariant functor. I have proven that indeed $OP(X)$ is a small category, so all I need to prove is that if $f$ is a continuous function from a topological space $X$ to $Y$, then $OP(f)$ should be a morphism (which in $\textbf{Cat}$ is a functor) from $OP(Y)$ to $OP(X)$. The problem is that the problem doesnt specify what $OP(-)$ does to the morphisms of $\textbf{Top}$. I dont know if there is a natural way to define what $OP(-)$ does to the arrows of $\textbf{Top}$. Any help would be greatly appreciated.

Thanks.

  • 0
    For $f\colon X \to Y$ take $OP(f)$ to be $f^{-1}$, the preimage. It takes open subsets of Y, i.e. objects in $OP(Y)$, to open subsets in X , i.e. objects in $OP(X)$, and preserves inclusion.2012-10-02
  • 0
    Thank you so much!! That is all I wanted to know. That $OP(f)=f^{-1}$. Sorry, but I am new to this subject, and rather bad at math, but is it supposed to be understood what $OP(-)$ does to the arrows of $\textbf{Top}$?2012-10-02
  • 0
    I don't know. As for catgeory theory, the slogan always is "take it to be the most obvious". [The Catsters](https://www.youtube.com/user/thecatsters) do a really good job at explaining category theoretical concepts, in my opninion.2012-10-02

1 Answers 1

1

Hint: By definition, $f\colon X\to Y$ is continuous iff for every open set $U\subseteq Y$ the set $f^{-1}(U)\subseteq X$ is open.