What is the definition of the curve?
a. Is the image of $x^2+y^2=1(x\neq0)$ a curve?
b. Is a point a curve?
Here are the definitions I found in Wikipedia that may help.
A curve is a topological space which is locally homeomorphic to a line.
Let I be an interval of real numbers (i.e. a non-empty connected subset of $R$). Then a curve is a continuous mapping $\gamma:I\to{}X$, where X is a topological space.
Two objects are homeomorphic if they can be deformed into each other by a continuous, invertible mapping. Right? But the second definition doesn't mention about the invertible.
Where can I get a rigorous definition of curve? Or in which topology textbook can I find the answer?