Intersection of Neighborhood of Diagonal with Inverse is Neighborhood
Jump to navigation
Jump to search
This article needs proofreading. Please check it for mathematical errors. If you believe there are none, please remove {{Proofread}} from the code.To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Proofread}} from the code. |
Theorem
Let $T = \struct{X, \tau}$ be a topological space.
Let $\tau_{X \times X}$ denote the product topology on the cartesian product $X \times X$.
Let $R$ be a neighborhood of the diagonal $\Delta_X$ of $X \times X$ in the product space $\struct{X \times X, \tau_{X \times X}}$.
Let $R^{-1}$ denote the inverse relation of $R$ where $R$ is viewed as a relation on $X \times X$.
Then:
- $R \cap R^{-1}$ is a neighborhood of $\Delta_X$ in $\struct{X \times X, \tau_{X \times X}}$.
Proof
From Set is Neighborhood of Subset iff Neighborhood of all Points of Subset:
- $\forall \tuple{x, x} \in \Delta_X : R$ is a neighborhood of $\tuple{x, x}$
From Inverse of Neighborhood of Diagonal Point is Neighborhood:
- $\forall \tuple{x, x} \in \Delta_X : R^{-1}$ is a neighborhood of $\tuple{x, x}$
From Intersection of Neighborhoods in Topological Space is Neighborhood
- $\forall \tuple{x, x} \in \Delta_X : R \cap R^{-1}$ is a neighborhood of $\tuple{x, x}$
From Set is Neighborhood of Subset iff Neighborhood of all Points of Subset:
- $R \cap R^{-1}$ is a neighborhood of $\Delta_X$ in $\struct{X \times X, \tau_{X \times X}}$
$\blacksquare$