I have been struggling with general topology and now, algebraic topology is simply murder. Some people seem to get on alright, but I am not one of them unfortunately.

*Please, the answer I need is ideally something very elaborate and extensive, possibly with easy-to-understand examples. Rewriting the definitions with concrete mathematical language and symbols won't help(those are readily accessible in my lecture notes). I want an explanation of what is happening and which part of the solid definitions are telling me so.*

One thing I would like to make clear is, I *know* the definitions, but I *don't understand* them; I can re-iterate them upon request, consult my lecture notes. Issue is I am writing down something I don't know what it *means*. It's like writing ancient Greek. I can remember the shapes of each character, their order and write some sentence down. But that's it. Can I explain it to someone with my *own words*? Break it down? Absolutely not. That is why I am here to ask someone who fully understands these ideas to do exactly that for me: break it down. Go in slow motion. Show me the moves and codes behind it.

Here's the definition I have for homotopy

A homotopy between maps $f,g:X \rightarrow Y$ is a map $h: X \times I \rightarrow Y$ such that

$$h(x,0)=f(x), h(x,1)=g(x) \in Y$$ where $x \in X$ and $I=[0,1]$. We say maps $f,g$ are homotopic.

It adds that,

A homotopy deforms the map $f$ continuously to $g$.

So, I am given two sets $X,Y$ whatever they are and two maps $f,g$ that takes an element of $X$ to an element of $Y$. I don't know if $f,g$ are bijective, only injective or surjective or whatever. No information on that. Just maps.

And this "homotopy" is a ... "map between maps"?
And even if so, what *exactly* is it doing? All the explanation seems to be done with this single line

$$h(x,0)=f(x), h(x,1)=g(x) \in Y$$

but no, I don't get what's happening. So $X$ has a bunch of elements $x$, and the product space $X \times I$ gives me elements of the form $\{x,t\} \in X \times I$. Okay. But this map $h$ qualifies as a homotopy as long as the above holds? Then why not just always define $h$ as $h(x,t)=f(x)$ for any $t \neq 1$ and $h(x,t)=g(x)$ for $t=1$, just as how we might define a piecewise function? Then I can define this "homotopy" on *any* maps.

Sure, it adds "deforms $f$ CONTINUOUSLY to $g$" but how is that stated in the definition itself? I see it nowhere.

Here is an example in my notes which didn't help me understand the definition,

Take $X=\{x\}$ the space with a single element $x$. Then a map $f:X \rightarrow Y$ is the same as and element $f(x) \in Y$. A homotopy $h: f \simeq g: X \rightarrow Y $ is the same as a path $h: I \rightarrow Y$ with initial point $h(0)=f(x)$ and terminal point $h(1)=g(x) \in Y$. A homotopy $h: f \simeq f: X \rightarrow Y$ is the same as a closed path $h: I \rightarrow Y$.

Well, first off what is a "path"? Intuition also doesn't make sense because when mapping one element to another, how can there be different "paths"? $x$ goes to $y$. Done. It's not like going from England to Singapore via either Amsterdam or Frankfurt (thus different paths) is it? Unless it's a map $X \rightarrow Z \rightarrow Y$ and telling me that $x \in X$ goes to $z_1 \in Z$ then to $y \in Y$ or $z_2 \in Z$ and then to $y \in Y$, that might be different paths from $x$ to $y$. But here, it is talking about $X$ and $Y$ only.

And why is this ignoring $x$? It says $h: I \rightarrow Y$? The problem I also have is, this is labelled "example" but it's not specific at all. "$h(0)=f(x)$ and $h(1)=g(x)$" over. So? What is this $h$ and how has it been defined? As a map, as a function of some sort? Maybe there are multiple homotopy thinkable, but then what are one or two of them?

It's like saying $f(1)=1$ and $f(2)=4$. Done. Well, to a newbie, maybe secondary school kids, it might be nice to give them an example be it linear $f(x)=ax+b$ or $f(x)=x^2$. To the eyes of the experienced, it might make perfect sense, a specific homotopy might pop out in their minds like popcorns but not in mine.

It's an utter nightmare. I know this is "abstract" math but can not some more specific-ness be put into it? Pictures and diagrams perhaps?

This is only the tip of the massive massive confusion and dumb-foundedness I am experiencing in this area of study. Maybe once it "clicks" it all goes down like an avalanche but so far it's nothing but counter-intuitive.

Can someone please make this possible for me to digest? Suggestions to good websites with examples and diagrams and extensive explanations are also welcome. Thank you