Wrap the real line around a circle by p(t)=exp(2πit). Every small arc has infinitely many separated copies upstairs, and each copy maps bijectively and continuously onto the arc. Locally the map loses no information; globally integer shifts have the same image. A covering map formalizes this local-sheet structure.
Predict
Predict whether every surjective local homeomorphism is a covering. Restrict the same exponential map to (0,2). It remains locally invertible and onto, but one point of the circle has only one preimage while nearby points have two. The missing domain endpoints prevent a uniform neighborhood from splitting into complete sheets.
Experiment
Compare fibers above a base point and nearby points.
Map
Fiber above 1
Nearby fiber size
R→S¹, exponential
All integers
Countably infinite
S¹→S¹, z↦z³
Three roots
Three
(0,2)→S¹, exponential
Only 1
Two
For the first map choose an argument arc (−1/8,1/8) modulo integers. Write three of its lifted intervals and explain why there are infinitely many more. In the last row, locate the partial intervals that run into missing endpoints.
Understand
A covering map p:E→B is a continuous surjection such that each b in B has an open neighborhood U whose full inverse image is a disjoint union of open sets V_j, each mapped homeomorphically onto U by p. Such a U is evenly covered, and the V_j are its sheets. The definition concerns the entire inverse image, not merely one local inverse near each preimage point.
For the exponential map, choose U as an arc represented by a real interval (a,b) with b−a<1. Its inverse image is the disjoint union of (a+k,b+k) over integers k. Restricted exponentiation has a continuous inverse argument on each interval, proving the covering condition. For z↦z^m, the same argument interval has m inverse branches with arguments divided by m and shifted by k/m. These produce m disjoint open arcs and exhaust the inverse image.
A covering is a local homeomorphism, since each preimage point lies in a sheet. The converse fails for the restricted exponential example. Fiber cardinality is locally constant under a genuine covering: over an evenly covered U, one point of each sheet lies above every base point. On a connected base, a locally constant cardinal cannot change, since each cardinal-value set and its complement are open. This proves the fiber-count obstruction without assuming a finite number of sheets.
The fiber has the discrete subspace topology: a sheet intersects it in exactly one point. This simple fact will make lifted endpoints rigid under a continuous deformation. Neither the total space nor the base is assumed simply connected by the definition. A cover may also have several connected components.
Apply
Check the trivial cover B×D→B when D is nonempty and discrete. Each B×a single point is a sheet over any open U in the base. If D is not discrete, this argument fails because the candidate sheets need not be open. The topology on the fiber factor is therefore part of the construction, not merely a counting choice.
Continue
When verifying a cover, exhibit an evenly covered neighborhood and every sheet above it. Local formulas alone can miss incomplete sheets. The next lesson uses compactness of the path parameter interval to assemble these local inverses into a unique lifted path from any specified starting point.
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
For p:S1→S1 given by p(z)=z^7, how many points lie above 1?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
A trivial cover is X×{1,2,3,4}→X. How many sheets lie over each evenly covered neighborhood?
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Prove p:R→S1, p(t)=exp(2πit), is a covering.
Structure your argument and preserve revisions
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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 z↦z^m is a covering for positive integer m.
Structure your argument and preserve revisions
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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.
Explain why exp(2πit):(0,2)→S1 is a surjective local homeomorphism but not a covering.
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 the number of sheets is constant over a connected base, allowing an infinite cardinal.
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.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
