Topology: distinguish hypotheses, models and invariants
Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 210 minutes, as an unvalidated planning estimate. This collection contains 12 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
Attempt the mixed problems before revealing guidance. State definitions, hypotheses and intermediate arguments, and identify any theorem you use without proving it.
These public tasks require human mathematical review. A saved response is not verified proof quality, a secure examination or evidence of universal course equivalence.
The suggested time is planning guidance; work is untimed. Preserve the initial attempt, consult targeted lessons, then revise your reasoning.
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0 of 12 tasks have recorded responses. No overall mastery score is assigned.
For dimension bound n=2, explain why the proof uses R⁵. In R⁴ take one triangle with vertices e₁,e₂,−e₁−e₂ and another with e₃,e₄,−e₃−e₄. Show their convex hulls meet despite disjoint vertices. Lift all vertices of the second triangle to fifth coordinate δ>0, leaving the first at zero, and prove the six lifted vertices are affinely independent. Separately explain why continuous injectivity alone does not prove embedding without compactness, using [0,2π) mapped onto the circle.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
In X=[0,3], collapse A=[0,1] to one point and B=[2,3] to another, leaving all other points separate. Prove the quotient is homeomorphic to [0,1] with an explicit map and a full inverse-continuity argument. Change only A to [0,1), retaining B=[2,3]. Determine whether the same formula has exactly the requested fibers and whether the quotient is T1. Construct a convergent sequence of equivalent pairs witnessing nonclosedness of the new relation. Submit an initial proof, a hypothesis audit, and a revised proof that explains the failed step; a reviewer response is evidence only if actually obtained.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
On X={p,a,b,c}, declare the open sets to be ∅ and every subset containing p. Verify the axioms, give a minimal basis, compute interior and closure for every kind of subset, and determine T0, T1, Hausdorff and the topology on Y={a,b,c}.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For R^N, construct a countable basis for the product topology. Prove the diagonal map d(t)=(t,t,…) is product-continuous but not box-continuous. Explain why a box-open set need not be a union of product-basic sets.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Identify real numbers whose difference is an integer. Prove R/Z is homeomorphic to the unit circle using an explicit open map. Explain why compactness of R cannot justify this step. Separately collapse all rational numbers in R to one point, leaving each irrational point separate. Prove this quotient is not T1 and any two nonempty open sets meet.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Give an uncountable set X the cocountable topology. Show every convergent sequence is eventually its limit. Nevertheless, for an infinite A⊆X and x∉A, describe precisely when x lies in the closure of A and construct a net from A converging to x in that case.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Prove a compact subset of a Hausdorff space is closed. Use this to prove the one-point compactification of the discrete integers Z is Hausdorff and compact, and give an explicit homeomorphism to a convergent sequence together with its limit.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Let C=({0}∪{1/n:n≥1})×[0,1]⊆R². Determine its connected components and path components. Prove it is not locally connected at any point of {0}×[0,1].
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Let I be any set and F a family of closed subsets of {0,1}^I in the product topology. Suppose every finite subfamily has nonempty intersection. Prove all constraints have a simultaneous solution. Explain where compactness and choice enter, and why testing a finite sample of constraints is not this hypothesis.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Let X be a normal T1 space and A,B disjoint closed subsets. Construct a continuous function into [−2,5] taking −2 on A and 5 on B. Explain how Tietze extends an arbitrary bounded continuous function on a closed subset. Does normal T1 alone imply metrizability? Supply a counterexample and prove the missing countability property fails.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Let αN be the one-point compactification of discrete N≥1 and K a compact Hausdorff space. Prove f:N→K extends continuously to αN exactly when the sequence f(n) converges in K. Apply this to f(n)=1/n into [0,1]. For its extension F to βN, prove that F vanishes exactly on βN minus N. Finally prove uniqueness of extensions into Hausdorff targets.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Prove that Q with its usual subspace topology cannot be given a complete compatible metric. Separately show its covering dimension is zero using clopen refinements of finite open covers. Explain why these conclusions are compatible, and why a topological embedding theorem does not imply closed image.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
For dimension bound n=2, explain why the proof uses R⁵. In R⁴ take one triangle with vertices e₁,e₂,−e₁−e₂ and another with e₃,e₄,−e₃−e₄. Show their convex hulls meet despite disjoint vertices. Lift all vertices of the second triangle to fifth coordinate δ>0, leaving the first at zero, and prove the six lifted vertices are affinely independent. Separately explain why continuous injectivity alone does not prove embedding without compactness, using [0,2π) mapped onto the circle.
In X=[0,3], collapse A=[0,1] to one point and B=[2,3] to another, leaving all other points separate. Prove the quotient is homeomorphic to [0,1] with an explicit map and a full inverse-continuity argument. Change only A to [0,1), retaining B=[2,3]. Determine whether the same formula has exactly the requested fibers and whether the quotient is T1. Construct a convergent sequence of equivalent pairs witnessing nonclosedness of the new relation. Submit an initial proof, a hypothesis audit, and a revised proof that explains the failed step; a reviewer response is evidence only if actually obtained.
On X={p,a,b,c}, declare the open sets to be ∅ and every subset containing p. Verify the axioms, give a minimal basis, compute interior and closure for every kind of subset, and determine T0, T1, Hausdorff and the topology on Y={a,b,c}.
For R^N, construct a countable basis for the product topology. Prove the diagonal map d(t)=(t,t,…) is product-continuous but not box-continuous. Explain why a box-open set need not be a union of product-basic sets.
Identify real numbers whose difference is an integer. Prove R/Z is homeomorphic to the unit circle using an explicit open map. Explain why compactness of R cannot justify this step. Separately collapse all rational numbers in R to one point, leaving each irrational point separate. Prove this quotient is not T1 and any two nonempty open sets meet.
Give an uncountable set X the cocountable topology. Show every convergent sequence is eventually its limit. Nevertheless, for an infinite A⊆X and x∉A, describe precisely when x lies in the closure of A and construct a net from A converging to x in that case.
Prove a compact subset of a Hausdorff space is closed. Use this to prove the one-point compactification of the discrete integers Z is Hausdorff and compact, and give an explicit homeomorphism to a convergent sequence together with its limit.
Let I be any set and F a family of closed subsets of {0,1}^I in the product topology. Suppose every finite subfamily has nonempty intersection. Prove all constraints have a simultaneous solution. Explain where compactness and choice enter, and why testing a finite sample of constraints is not this hypothesis.
Let X be a normal T1 space and A,B disjoint closed subsets. Construct a continuous function into [−2,5] taking −2 on A and 5 on B. Explain how Tietze extends an arbitrary bounded continuous function on a closed subset. Does normal T1 alone imply metrizability? Supply a counterexample and prove the missing countability property fails.
Let αN be the one-point compactification of discrete N≥1 and K a compact Hausdorff space. Prove f:N→K extends continuously to αN exactly when the sequence f(n) converges in K. Apply this to f(n)=1/n into [0,1]. For its extension F to βN, prove that F vanishes exactly on βN minus N. Finally prove uniqueness of extensions into Hausdorff targets.
Prove that Q with its usual subspace topology cannot be given a complete compatible metric. Separately show its covering dimension is zero using clopen refinements of finite open covers. Explain why these conclusions are compatible, and why a topological embedding theorem does not imply closed image.
