f is homotopic to g

If you have X and then take I
And cross it with this space
And there's an H that goes to Y
To put your things in place

Then H of x at time t one
Will send you along g
Where you had f at time t naught
And deformed continuously.

The manifold of Venice

And lo, i am Hausdorff, compact and simply connected;
My neighborhoods are homeomorphic to Euclidian space;
But i am wont for substance.

Embed me, triangulate me, puncture me; am i not thus deformed?
Circles, circles round me wending, to search the answer i yearn.

But alas, fundamentally i am trivial.