Topological manifold example

Question

$\theta(x,x^2)=x$

$\Bbb X =${$(x,x^2)| x$ in $\Bbb R$}

And V is subset of $\Bbb R$

$dim\Bbb X=1$

My instructor said that this is topological manifold.

Why?

Please can you explain me? This example is from my notebook. But there is No explanation except for this. Thank you for help.

Answer 1

We know that : The Graph of a smooth function is a manifolds.
For a subset of $A ⊂ \mathbb R^n$ and a function $f : A→\mathbb R^m$, the graph of $f$ is defined to be the subset $$G( f ) = \{(x, f (x)) ∈ A×\mathbb R^m\}$$ If $U $is an open subset of $\mathbb R^n$ and $f : U →\mathbb R^n$ is $C^{\infty}$, then the two maps $$f : G( f )→U$$ $$(x, f (x)) \mapsto x$$ and $$(1, f ) : U →G( f )$$$$ x \mapsto (x, f (x))$$ are continuous and inverse to each other, and so are homeomorphisms. The graph $G( f )$ of a $C^{\infty}$ function $f : U →\mathbb R^m$ has an atlas with a single chart $(G( f ),f )$, and is therefore a $C^{\infty}$ manifold. This shows that many of the familiar surfaces of calculus, for example your question are maniflods .

Answer 2

Check that $\theta:X \to\mathbb{R}$ it's an homeomorphism, where $\mathbb{R}$ has the euclidean topology; then you conclude since the codomain is surely a one-dimensional topological manifold.