Courses / Mathematics / Point-set topology: spaces, constructions and separation Mathematics · unit version 1 Point-set topology: spaces, constructions and separation Twenty-eight proof-centered lessons on topological constructions, convergence, separation, compactness, connectedness, extension, compactification, category and introductory dimension. Choice principles are explicit in the ultrafilter and product arguments. Metrization and Stone–Čech are developed through stated construction lemmas; the compact-metric finite-dimensional embedding theorem is proved through general position and Baire; broader noncompact forms remain separate. Written work requires mathematical review; this is not independently assessed course equivalence.
Open any lesson. Completion records participation; automatic practice checks and human judgment of written arguments are separate. This unit retains its own versioned requirements.
Readiness and preparation Check your starting point Try these short questions on paper, then compare your reasoning. They are optional self-checks, not a placement score or a navigation requirement. Use a targeted review when useful and return with your browser’s Back control.
How do you prove that two sets are equal? Topology repeatedly compares families of subsets and inverse images.
Compare reasoning and find a refresher Prove both inclusions: an arbitrary member of the first belongs to the second, and conversely. A diagram or a finite sample does not establish arbitrary membership.
Review set equality arguments
Does taking an inverse image preserve arbitrary unions and finite intersections? Continuity and generated topologies depend on these set identities.
Compare reasoning and find a refresher Yes. Membership in the inverse image means that the function value lies in the target set; applying “some” for a union and “all” for an intersection proves the identities. No inverse function is required.
Review inverse images
What is an open neighborhood in a metric space? Metric examples provide intuition before topology is defined without distance.
Compare reasoning and find a refresher An open neighborhood of x is an open set containing x; it contains some ball centered at x. The radius may depend on the point. General topological spaces use open sets as primary data instead of assuming a distance.
Review metric neighborhoods
Loading saved mathematics work…
Topology: distinguish hypotheses, models and invariants
The mixed assessment has separate response history and review criteria; it does not change lesson participation records.
Lesson sequence Each linked lesson opens on its own page. Expand its details here when you want to check the objectives before choosing.
Lesson details and objectives Check the three topology axioms on a finite family.
Check the three topology axioms on a finite family. Distinguish arbitrary unions from finite intersections. Compare coarser and finer topologies on the same set. 6 practice tasks.
Lesson details and objectives Verify the basis covering and local-intersection conditions.
Verify the basis covering and local-intersection conditions. Prove that unions of basis elements form a topology. Use basis refinement to compare topologies. 6 practice tasks.
Lesson details and objectives Generate a basis from finite subbasic intersections.
Generate a basis from finite subbasic intersections. Construct the coarsest topology making given maps continuous. Prove the initial-topology mapping criterion. 6 practice tasks.
Lesson details and objectives Construct the topology inherited by a subset.
Construct the topology inherited by a subset. Distinguish relative openness from ambient openness. Prove continuity and closure statements for subspaces. 6 practice tasks.
Lesson details and objectives Compute closure and interior from neighborhood definitions.
Compute closure and interior from neighborhood definitions. Prove closure laws without relying on sequences. Track boundary and closure through a subspace. 6 practice tasks.
Lesson details and objectives Relate global open-preimage continuity to the pointwise neighborhood condition.
Relate global open-preimage continuity to the pointwise neighborhood condition. Prove the closed-set and closure criteria for continuity. Use finite closed covers to glue continuous maps. 6 practice tasks.
Lesson details and objectives Distinguish a continuous bijection from a homeomorphism.
Distinguish a continuous bijection from a homeomorphism. Prove equivalent open-map and closed-map tests for bijections. Use a preserved property to rule out a homeomorphism. 6 practice tasks.
Lesson details and objectives Construct the product topology from open rectangles.
Construct the product topology from open rectangles. Prove continuity of a map into a product by its coordinates. Distinguish open projections from closed projections. 6 practice tasks.
Lesson details and objectives Describe product neighborhoods with finitely many restricted coordinates.
Describe product neighborhoods with finitely many restricted coordinates. Prove the infinite-product coordinate continuity criterion. Construct a sequence separating product and box convergence. 6 practice tasks.
Lesson details and objectives Define quotient openness through full inverse images.
Define quotient openness through full inverse images. Prove the mapping-out universal property. Recognize open or closed continuous surjections as quotient maps. 6 practice tasks.
Lesson details and objectives Compute saturated sets as unions of equivalence classes.
Compute saturated sets as unions of equivalence classes. Construct maps out of quotients by checking constancy on fibers. Analyze endpoint identification and a non-Hausdorff gluing example. 8 practice tasks.
Lesson details and objectives Separate the three point-separation axioms.
Separate the three point-separation axioms. Prove the closed-singleton characterization of T₁. Prove uniqueness of limits in Hausdorff spaces. 6 practice tasks.
Lesson details and objectives State point–closed-set and closed-set separation with an explicit T₁ convention.
State point–closed-set and closed-set separation with an explicit T₁ convention. Prove the neighborhood-shrinking characterizations. Verify metric normality and the limits of inherited separation. 6 practice tasks.
Lesson details and objectives Distinguish first countability, second countability and separability.
Distinguish first countability, second countability and separability. Prove second countability implies separability and Lindelöfness. Use the lower-limit line to refute a converse. 6 practice tasks.
Lesson details and objectives Prove closure can be tested by sequences in first-countable spaces.
Prove closure can be tested by sequences in first-countable spaces. Identify why the proof fails without a countable local base. Construct a sequentially continuous map that is not continuous. 6 practice tasks.
Lesson details and objectives Define directed sets and eventual membership for nets.
Define directed sets and eventual membership for nets. Prove the net characterization of closure and continuity. Characterize Hausdorffness by uniqueness of net limits. 6 practice tasks.
Lesson details and objectives Prove the finite-intersection characterization of compactness.
Prove the finite-intersection characterization of compactness. Distinguish compactness from a metric boundedness condition. Use explicit covers to prove failures of compactness. 6 practice tasks.
Lesson details and objectives Prove continuous images and closed subspaces of compact spaces are compact.
Prove continuous images and closed subspaces of compact spaces are compact. Prove compact subsets of Hausdorff spaces are closed. Derive the compact-to-Hausdorff homeomorphism criterion. 6 practice tasks.
Lesson details and objectives Use compact neighborhoods with the Hausdorff hypothesis explicit.
Use compact neighborhoods with the Hausdorff hypothesis explicit. Construct and prove compactness of the one-point compactification. Show local compactness supplies separation from the added point. 6 practice tasks.
Lesson details and objectives Prove connectedness is preserved by continuous images and closure.
Prove connectedness is preserved by continuous images and closure. Construct components as maximal connected subsets. Distinguish components from open pieces without local connectedness. 6 practice tasks.
Lesson details and objectives Prove path connectedness implies connectedness.
Prove path connectedness implies connectedness. Compare connected and path components in locally path-connected spaces. Analyze the topologist’s sine curve as a genuine counterexample. 6 practice tasks.
Lesson details and objectives Prove the ultrafilter characterization of compactness.
Prove the ultrafilter characterization of compactness. Derive product compactness from coordinate filter limits. State the maximal-principle and choice dependencies explicitly. 6 practice tasks.
Lesson details and objectives Construct a continuous separator by nested dyadic neighborhoods.
Construct a continuous separator by nested dyadic neighborhoods. Prove regular second-countable spaces are normal under the T₁ convention. Explain the countable-function embedding that yields a compatible metric. 6 practice tasks.
Lesson details and objectives Construct a bounded extension using Urysohn separators and geometric residual control.
Construct a bounded extension using Urysohn separators and geometric residual control. Prove continuity of the uniformly convergent correction series. Explain the closed-subspace and normality hypotheses. 6 practice tasks.
Lesson details and objectives Define a compactification as a dense embedding into a compact Hausdorff space.
Define a compactification as a dense embedding into a compact Hausdorff space. Construct Stone–Čech through all bounded continuous coordinate functions. Prove its extension property and distinguish it from one-point compactification. 6 practice tasks.
Lesson details and objectives Prove the complete-metric Baire theorem with nested closed balls.
Prove the complete-metric Baire theorem with nested closed balls. Separate meagreness from measure and cardinality. Apply category reasoning to a uniform boundedness argument. 6 practice tasks.
Lesson details and objectives Define covering dimension by finite open refinements and multiplicity.
Define covering dimension by finite open refinements and multiplicity. Prove dimension-zero and interval examples with explicit cover arguments. Prove the compact-metric finite-dimensional embedding theorem using general position and Baire. 8 practice tasks.
Lesson details and objectives Select invariants and constructions appropriate to a new comparison.
Select invariants and constructions appropriate to a new comparison. Audit hypotheses across product, quotient and compactness arguments. Write a proof portfolio distinguishing full proofs from theorem dependencies. 6 practice tasks.
