Neighbourhood, Open Sets, Closed Sets
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Introduction to Neighbourhoods of a Point
In real analysis, the concept of Neighbourhoods of a Point is essential for understanding open and closed sets. This article delves into the definitions, illustrative examples, and fundamental theorems associated with neighbourhoods and deleted neighbourhoods. By incorporating standard mathematical notation such as \(\mathbb{R}\) for the set of real numbers, this guide aims to enhance your comprehension while ensuring the content is optimized for search engines.
Neighbourhood of a Point
Definition: [Neighbourhood]
Let \(S\subseteq \mathbb{R}\) and \(\alpha \in \mathbb{R}\). \(S\) is said to be a neighbourhood of \(\alpha \) if there exits an open interval \( I \) such that \[\alpha\in I\subseteq S .\]
For convenience, we will take an open interval \( (\alpha – \delta, \alpha + \delta ) \) provide \( \delta \gt 0 \) as a neighbourhood of \(\alpha \) and is denoted by \( \operatorname{N}_{\delta}(\alpha) \). Therefore \[\operatorname{N}_{\delta}(\alpha)=(\alpha – \delta, \alpha + \delta )=\set{x\in \mathbb{R}: |x-\alpha|\lt \delta }.\]
Deleted Neighbourhood of a Point
Definition: [Deleted Neighbourhood]
Let \(S\subseteq \mathbb{R}\) and \(\alpha \in \mathbb{R}\). \(S\) is said to be a deleted neighbourhood of \(\alpha \) if \[S=(\alpha – \delta, \alpha + \delta )-\set{\alpha}\]
And is denoted by \( \operatorname{N}^{\prime}_{\delta}(\alpha) \). Therefore \[\operatorname{N}^{\prime}_{\delta}(\alpha)=(\alpha – \delta, \alpha + \delta )-\set{\alpha}=\set{x\in \mathbb{R}: 0\lt |x -\alpha|\lt \delta }.\]
Theorems
Theorem-1
Statement:
Let \(\alpha \in \mathbb{R} \) and \(0\lt \delta_{1} \leq \delta_{2} \) then \[\operatorname{N}_{\delta_{1}}(\alpha)\subseteq \operatorname{N}_{\delta_{2}}(\alpha).\]
Proof:
We have \(\alpha \in \mathbb{R} \) and \(0\lt \delta_{1} \leq \delta_{2} \).
To prove \(\operatorname{N}_{\delta_{1}}(\alpha)\subseteq \operatorname{N}_{\delta_{2}}(\alpha)\)
Let \[\begin{align} & \beta \in \operatorname{N}_{\delta_{1}}(\alpha) \nonumber \\ \implies & |\beta -\alpha|\lt \delta_{1} \nonumber \\ \implies & |\beta -\alpha|\lt \delta_{2} \nonumber \\ \implies & \beta \in \operatorname{N}_{\delta_{2}}(\alpha) \nonumber \end{align}\]
Therefore \(\operatorname{N}_{\delta_{1}}(\alpha)\subseteq \operatorname{N}_{\delta_{2}}(\alpha)\).
Conclusion
In conclusion, the study of Neighbourhoods of a Point is a fundamental aspect of real analysis, laying the groundwork for understanding more complex topics such as continuity, limits, and topological spaces. Through precise definitions and rigorous proofs, we gain valuable insights into how points and their surroundings interact within the realm of mathematics. This exploration reinforces the importance of clear mathematical notation and logical reasoning in forming a robust analytical framework.
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