Open Sets
\(\epsilon\)-neighborhood of \(a\)
Given \(a \in \mathbf{R}\) and \(\epsilon > 0\), the \(\epsilon\)-neighborhood of \(a\) is the set
$$
\begin{equation*}
V_{\epsilon}(a) = \{x \in \mathbf{R}: |x - a| < \epsilon\}.
\end{equation*}
$$
In other words, \(V_{\epsilon}(a)\) is the open interval \((a - \epsilon, a + \epsilon)\), centered at \(a\) with radius \(\epsilon\).
Open Sets
Based on the previous defintion, we can now define what it means for a set to be open
(3.2.1) A set \(O \in \mathbf{R}\) is open if for all points \(a \in O\), there exists an \(\epsilon\)-neighborhood \(V_{\epsilon}(a) \subseteq O\).
What does this mean? Consider the set \((c,d)=\{x \in \mathbf{R}, c < x < d\}\). \((c,d)\) is an open set. To see why, let \(x \in (c,d)\) be arbitrary. Set \(\epsilon = \min\{x - c, d - x\}\), then from this we see that \(V_{\epsilon}(a) \subseteq (c,d)\). For example if you take \((1,2)\) and pick \(x\) to be 1.7, we will then set \(\epsilon\) to be 0.3 which is what we need since \(1.7\pm \epsilon\) will always be
Other Definitions and Properties
- Limit Points
- Closed Sets
- Compact Sets
- Closure
- Absolute Value Function
- Sequences, Subsequences and Convergence
- Series and Series Convergence
- Show the Limit Template
References: