PTOP:010117Cofinite opens force infinite sets dense and kill Hausdorff separationCofinite Topology AxiomsCofinite Closure Compactness SeparationPrivateLabels unavailablePrivate saving unavailableTopological spaces, bases, subbases, and neighborhood systemsPoint-Set Topology · Mixed reviewProgress not loaded
PTOP:010118Dictionary order and the plane subspace topology are incomparableOrder Topology Basis ComparisonDictionary Order Subspace MismatchPrivateLabels unavailablePrivate saving unavailableTopological spaces, bases, subbases, and neighborhood systemsPoint-Set Topology · Mixed reviewProgress not loaded
PTOP:010219The diagonal breaks in the box topology, but addition survivesBox Versus Product TopologyDiagonal Failure Box Addition Continuity AuditPrivateLabels unavailablePrivate saving unavailableContinuity, initial topologies, products, and netsPoint-Set Topology · Mixed reviewProgress not loaded
PTOP:010220Finite closed pasting works; a countable closed cover can failFinite Closed Pasting LemmaInfinite Closed Cover Pasting FailurePrivateLabels unavailablePrivate saving unavailableContinuity, initial topologies, products, and netsPoint-Set Topology · Mixed reviewProgress not loaded
PTOP:010321A quotient of a compact Hausdorff space is Hausdorff exactly when the relation is closedClosed Relation Quotient HausdorffEndpoint Identification CirclePrivateLabels unavailablePrivate saving unavailableQuotient spaces, identification maps, and universal descentPoint-Set Topology · Mixed reviewProgress not loaded
PTOP:010422Regular plus Lindelöf yields normality by countable shrinkingRegular Lindelof Normal ProofCountable Shrinking OF CoversPrivateLabels unavailablePrivate saving unavailableSeparation and countability axiomsPoint-Set Topology · Mixed reviewProgress not loaded
PTOP:010523One-point compactification is Hausdorff exactly for locally compact Hausdorff spacesOne Point Compactification TopologyLocal Compactness Hausdorff CriterionPrivateLabels unavailablePrivate saving unavailableCompactness, local compactness, and compactificationPoint-Set Topology · Mixed reviewProgress not loaded
PTOP:010524Compact Hausdorff spaces are normal by closed-neighborhood shrinkingCompact Hausdorff RegularityCompact Hausdorff NormalityPrivateLabels unavailablePrivate saving unavailableCompactness, local compactness, and compactificationPoint-Set Topology · Mixed reviewProgress not loaded
PTOP:010525First countability turns compact cluster points into subsequencesCompact First Countable SequentialProduct Compact Not Sequentially CompactPrivateLabels unavailablePrivate saving unavailableCompactness, local compactness, and compactificationPoint-Set Topology · Mixed reviewProgress not loaded
PTOP:010626The topologist's comb is path connected yet not locally connectedPath Connectedness OF CombLocal Connectedness Failure AT Limit ToothPrivateLabels unavailablePrivate saving unavailableConnectedness, path connectedness, and componentsPoint-Set Topology · Mixed reviewProgress not loaded