Search found 284 matches

by Tsakanikas Nickos
Fri Nov 13, 2015 12:39 am
Forum: Category theory
Topic: Elementary Category Theory - 1
Replies: 0
Views: 2855

Elementary Category Theory - 1

Let \( \mathcal{F} : \mathcal{C} \longrightarrow \mathcal{D} \) be a functor between the categories \( \mathcal{C} \) and \( \mathcal{D} \). Show that \( \mathcal{F} \) is an equivalence if and only if \( \mathcal{F} \) induces bijections on the morphism sets and, additionally, for every object \( \...
by Tsakanikas Nickos
Fri Nov 13, 2015 12:37 am
Forum: Complex Analysis
Topic: Are These Sets Biholomorphic?
Replies: 2
Views: 3132

Are These Sets Biholomorphic?

Is the unit disc \( \displaystyle \mathbb{D} = \left\{ z \in \mathbb{C} \, \Big| \, |z| < 1 \right\} \) biholomorphic to \( \mathbb{C} \)? Is the punctured unit disc \( \displaystyle \mathbb{D}^{*} = \left\{ z \in \mathbb{C} \, \Big| \, 0< |z| < 1 \right\} \) biholomorphic to \( \mathbb{C} \smallse...
by Tsakanikas Nickos
Tue Nov 10, 2015 11:07 pm
Forum: Real Analysis
Topic: Proper Mappings
Replies: 0
Views: 1472

Proper Mappings

Definition : A continuous mapping \( \displaystyle f : X \longrightarrow Y \) between two locally compact topological spaces is called proper if the inverse image of every compact subset of \( \displaystyle Y \) under \( \displaystyle f \) is a compact subset of \( \displaystyle X \). 1. Give an ex...
by Tsakanikas Nickos
Tue Nov 10, 2015 9:28 pm
Forum: Complex Analysis
Topic: Open And Discrete
Replies: 0
Views: 2053

Open And Discrete

Let \( \displaystyle f \) be a non-constant holomorphic function defined on a region \( \displaystyle \Omega \) of \( \mathbb{C} \). Show that \( \displaystyle f \) is open and discrete. Note that "discrete" means that for all \( \displaystyle y \in \mathbb{C} \, , \, f^{-1} \left( \{y \} ...