A non-compact space in topology is a type of space that does not satisfy the properties of compactness. A space is compact if every open cover has a finite subcover, meaning that from any collection of open sets that cover the space, you can select a finite number of them that still cover the entire space. Non-compact spaces can exhibit behaviors like having infinite open covers without a finite subcover. Common examples of non-compact spaces include the real numbers ℝ and the open interval (0, 1). These spaces are not compact because they can be covered by open sets that cannot be reduced to a finite number. Understanding non-compact spaces is essential in various areas of mathematics, including analysis and topology.