# Topology, lemiscate not an embedded submanifold

1. Mar 30, 2013

### BrainHurts

1. The problem statement, all variables and given/known data

Show that the image of the curve Let β: (-π,π) → ℝ2 be given by β(t) = (sin2t, sint)
is not an embedded submanifold of ℝ2

2. Relevant equations

3. The attempt at a solution

So I'm not too great with the topology. I do see that β'(t) = (2cos2t, cost) ≠ 0 for all t. So β is a regular curve and as a result β is an immersion.

So here's what I have to work with.

A subset S of M is an embedded submanifold of M if:
- S is a smooth Manifold ( the image of (-π,π) under β is smooth)
-the inclusion map S to M is a smooth embedding

So this means that the topology of S (in our case β(-π,π)) is the subspace topology coming from M (in our case ℝ2)

We declare a set U$\subset$β(-π,π) is open if and only if there exists a set W$\subset$ℝ2 (namely the open discs) such that

U = W $\cap$ S

So to be more specific let's take U = β(-π/6, π/6), this set includes the origin.

Let W be the open unit disc B(0,1) which includes the origin. I've done a lot of reading where if we remove the origin we have trouble, not really seeing how W $\cap$ S will look like the letter X. Any thoughts?

Last edited: Mar 30, 2013