[Verse 1]
Start with sets and open neighborhoods
Topology begins where distance fades
Continuity through inverse images
Compact sets where every cover's made
Connected spaces can't be torn apart
Separation axioms set the rules
Hausdorff spaces keep their points distinct
These are topology's fundamental tools
[Chorus]
Bend and stretch but never tear
Topology's the shape we share
From point-set theory to homology
Geometry sets our spirits free
Manifolds and bundles too
Curvature in all we do
This is math's most visual art
Topology and geometry's heart
[Verse 2]
Fundamental groups track loops around
Covering spaces lift the paths we take
Van Kampen joins the pieces with a sound
Algebraic topology's first breakthrough break
Then homology counts the holes inside
Cohomology looks the other way
Exact sequences where the kernels hide
Mayer-Vietoris splits what won't decay
[Chorus]
Bend and stretch but never tear
Topology's the shape we share
From point-set theory to homology
Geometry sets our spirits free
Manifolds and bundles too
Curvature in all we do
This is math's most visual art
Topology and geometry's heart
[Verse 3]
Smooth manifolds where calculus lives on
Tangent bundles hold velocity's dream
Vector fields point where motion's drawn
Differential forms integrate the scheme
Riemannian metrics measure length and angle
Connections parallel transport the way
Curvature shows how space can mangle
Geodesics find the shortest ray
[Bridge]
Varieties algebraic, affine and projective
Schemes generalize with functors collective
Knots twist through three-dimensional space
Jones polynomial gives invariant grace
Symplectic structures preserve the flow
Hamiltonian mechanics makes it go
[Verse 4]
Higher homotopy groups extend the tale
Fibrations give us long exact rows
Spectral sequences when computations fail
Contact geometry where odd dimension glows
Floer homology bridges topology's gap
With analysis in infinite dimensions
Sheaf cohomology fills the theoretical map
Riemann-Roch counts with careful intentions
[Chorus]
Bend and stretch but never tear
Topology's the shape we share
From point-set theory to homology
Geometry sets our spirits free
Manifolds and bundles too
Curvature in all we do
This is math's most visual art
Topology and geometry's heart
[Outro]
From simple open sets to quantum fields
The shapes of space are what topology yields
In every curve and every twisted line
Mathematics shows its grand design