Contracting a contractible set in $\mathbb R^2$

This is a special case of a more general theorem due to Moore, whose proof should be in the book R.Wilder, "Topology of manifolds". Dimension 2 is very different from higher dimensions (2-dimensional homology manifolds are topological manifolds). Bing (I think) proved that contracting a wild arc in $R^3$ results in a space which is not a manifold.

There is a bijection between irreducible components of the generic fiber and irreducible components passing through it.

Is there any proof for this formula $\lim_{n \to \infty} \prod_{k=1}^n \left (1+\frac {kx}{n^2} \right) =e^{x/2}$?

Prove that $\mathbb{N}$ with cofinite topology is not path-connected space.

How to calculate the limit of $\frac{a_n}{n^2}$ for the sequence $a_{n+1}=a_n+\frac{2 a_{n-1}}{n+1}$?

Does $\lim_{(x,y)\to(0,0)}[x\sin (1/y)+y\sin (1/x)]$ exist?

Good book resources (not websites) to learn number theory on my own? [duplicate]

Find $\sum\limits_{k=0}^{n-1}\,\omega_n^{k^2\ell}$, where $\omega_n:=\exp\left(\frac{2\pi\text{i}}{n}\right)$.

Is there a (foundational) type theory with the features I'm looking for?

Is this proof that all metric spaces are Hausdorff spaces correct?

Smooth approximation of absolute value inequalities

Is the punctured plane homotopy equivalent to the circle?

How to find orthonormal frame of given metric?