Having moved through the preceding compactness ideas, we now study sequential forms of compactness. The focus keyword for this lesson is sequential forms of 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[Sequential Compactness] A metric space $X$ is $sequentially compact$ if every sequence in $X$ has a subsequence converging to a point of $X$. ::: :::definition[Limit-Point Compactness] A metric space $X$ has the $limit-point property$ if every infinite subset of $X$ has a limit point in $X$. ::: :::theorem[Sequential Characterisation of Compactness] A metric space is compact if and only if every sequence in it has a convergent subsequence. ::: :::proof Given that $X$ is a metric space. To prove the sequential characterisation. Compactness implies completeness and total boundedness, hence every sequence has a convergent subsequence. Conversely, if every sequence has a convergent subsequence, then every infinite subset has a limit point, and in metric spaces this implies compactness. $\square$ ::: :::theorem[Equivalent Characterisations of Compactness] Compactness, completeness plus total boundedness, the limit-point property, and sequential compactness are equivalent for metric spaces. ::: :::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 Subspaces and Countable Covers] 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-subspaces-and-countable-covers button-ghost: Previous Lesson | /functional-analysis/metric-spaces/compactness-and-its-consequences-in-metric-spaces/completeness-and-compactness :::