Having moved through the preceding compactness ideas, we now study local compactness. The focus keyword for this lesson is local compactness. This page records the definitions, theorems, proofs, examples, and practice items needed for undergraduate work in compactness and its consequences in metric spaces. :::definition[Locally Compact Metric Space] A metric space $(X,d)$ is $locally compact$ if for every $x\in X$ there exists $r>0$ such that $\overline{S(x,r)}$ is compact. ::: :::example[Real Line Is Locally Compact but Not Compact] In $\mathbb{R}$, the closure of a small ball is a closed bounded interval, hence compact. But $\mathbb{R}$ itself is not compact. ::: :::example[Infinite Discrete Spaces Are Locally Compact] Every singleton in a discrete metric space is compact, so an infinite discrete metric space is locally compact but not compact. ::: :::example[Compact Spaces Are Locally Compact] Every compact metric space is locally compact because closed subsets of compact metric spaces are compact. ::: :::solved-problem A continuous open image of a locally compact metric space is locally compact. ::: :::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 Compact Neighbourhoods and Exhaustions] 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/compact-neighbourhoods-and-exhaustions button-ghost: Previous Lesson | /functional-analysis/metric-spaces/compactness-and-its-consequences-in-metric-spaces/uniform-continuity-on-compact-domains :::