Having moved through the preceding compactness ideas, we now study compact neighbourhoods and exhaustions. The focus keyword for this lesson is compact neighbourhoods and exhaustions. This page records the definitions, theorems, proofs, examples, and practice items needed for undergraduate work in compactness and its consequences in metric spaces. :::definition[Open Neighbourhood of a Set] For a nonempty subset $A$ of $(X,d)$ and $r>0$, define $$\begin{equation*} V_r(A)=\{x\in X:d(x,A)<r\}. \end{equation*}$$ This is the $open r-neighbourhood$ of $A$. ::: :::theorem[Openness of Set Neighbourhoods] The set $V_r(A)$ is open and contains $A$. ::: :::proof Given that $x\in V_r(A)$. To prove openness. Choose $\eta>0$ with $d(x,A)+\eta<r$. If $d(x,y)<\eta$, then $d(y,A)<r$. Hence a ball around $x$ lies in $V_r(A)$. $\square$ ::: :::theorem[Compact Neighbourhood of a Compact Set] A compact subset of a locally compact metric space has an open neighbourhood with compact closure. ::: :::theorem[Sigma-Compactness, Separability, and Compact Exhaustions] In locally compact metric spaces, compact exhaustions, sigma-compactness, and separability are closely connected through countable compact neighbourhood constructions. ::: :::exercise [1] State the main definition or theorem from this lesson. [2] Give one example where the hypothesis is used. [3] Explain one common mistake related to this topic. ::: :::answer [1] See the first formal block of the lesson. [2] The examples in the lesson provide standard choices. [3] A common mistake is to apply a Euclidean compactness rule in an arbitrary metric space. ::: :::faq[Frequently Asked Questions] Q: Why is the metric important? A: Compactness, boundedness, convergence, and openness are all interpreted using the chosen metric. Q: What should students check first? A: Identify the space, the metric, and the exact compactness criterion being used. ::: :::call-to-action[Continue to Equicontinuous Families in Function Spaces] subtitle: Review the definitions, proofs, and examples before moving to the next compactness idea. button: Next Lesson | /functional-analysis/metric-spaces/compactness-and-its-consequences-in-metric-spaces/equicontinuous-families-in-function-spaces button-ghost: Previous Lesson | /functional-analysis/metric-spaces/compactness-and-its-consequences-in-metric-spaces/local-compactness :::