I am currently studying undergraduate algebraic topology. We have in our notes the definition of homotopy of continuous maps:
Two continuous maps $\phi_0,\phi_1:X \rightarrow Y$ are said to be homotopic if there exists a continuous map $F:X \times I \rightarrow Y$ (where $X \times I$ has the product topology) such that
$$F(x,0) = \phi_0(x), \; F(x,1) = \phi_1(x), \; \forall x \in X$$
This definition makes sense to me. I am now trying to gain a more intuitive understanding.
For comparison, we have previously had the definition of based homotopy of paths in a topological space $X$. This is relatively easy to understand - two paths with the same end points are homotopic if you can deform one into the other in a continuous manner.
I'm getting mixed up, though, over how a continuous map, say $\phi_0$, from a space $X$ into a space $Y$ can be homotopic in such a sense to another such map, $\phi_1$. If things have a Euclidean topology, what sort of continuous maps are not homotopic?
If for example, $X = Y = T^2, \; \phi_0 = id_x$, what might be some examples of continuous maps $\phi$ which $\phi_0$ is or is not homotopic to?
Is there a nice way to visualise this, in a similar manner to how we can visualise based homotopy of paths?