Search found 284 matches
- Fri Nov 13, 2015 12:39 am
- Forum: Category theory
- Topic: Elementary Category Theory - 1
- Replies: 0
- Views: 3625
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 \( \...
- Fri Nov 13, 2015 12:37 am
- Forum: Complex Analysis
- Topic: Are These Sets Biholomorphic?
- Replies: 2
- Views: 3156
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...
- Tue Nov 10, 2015 11:07 pm
- Forum: Real Analysis
- Topic: Proper Mappings
- Replies: 0
- Views: 1486
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...
- Tue Nov 10, 2015 9:28 pm
- Forum: Complex Analysis
- Topic: Open And Discrete
- Replies: 0
- Views: 2118
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 \} ...