\(\newcommand{\from}{\colon}\)\(\newcommand{\cross}{\times}\)\(\newcommand{\cover}[1]{{\widetilde{#1}}}\)Questions
handed in by a student on 2017-11-07.

Lecture -

1. What is the "Y" in the definition of HLP? Does it have any restriction?

Answer: \(Y\) can be any topological space.  This is the reason that the
proof of the HLP for covering maps has one particularly subtle step.
It occurs in Hatcher's book on page 30, line -7, in the sentence
"After replacing N by a smaller neighborhood..."

For a fuller discussion see:

https://math.stackexchange.com/questions/1290032/homotopy-lifting-property-of-mathbbr-to-s1-in-hatcher

That posting is also a lesson in asking questions; sometimes you will
receive a truly informative answer.

General - 

1. What is the difference between homomorphism, homeomorphism,
isomorphism? (It's a mess for me).

Answer: Let's discuss the algebraic ones first, in the context of
groups.  So, suppose that \(G\) and \(H\) are groups.

Definition: A function \(\phi \from G \to H\) is a _homomorphism_ if for
all \(g, h \in G\) we have \(\phi(gh) = \phi(g)\phi(h)\).

Definition: A homomorphism \(\phi \from G \to H\) is an _isomorphism_ if
it is bijective and its inverse is a homomorphism.

Now for the topological versions. Suppose that \(X\) and \(Y\) are
topological spaces.

Definition: A function \(f \from X \to Y\) is a _map_ if it is
continuous.

Definition: A map \(f \from X \to Y\) is a _homeomorphism_ if it is
bijective and its inverse is also a map.

You may have noticed that there is a sort of dictionary here:

Algebra          Topology
-------          --------
group            space
subgroup         subspace
homomorphism     map
isomorphism      homeomorphism

As an exercise, you should think about the corresponding column of
notions for linear algebra.  We could add futher columns still: real
analysis, rings, fields, complex analysis, algebraic geometry, and so
on.  In fact, most areas of mathematics have some notion of "morphism"
and of "isomorphism".  Sometimes there is more than one sensible
notion of morphism or isomorphism, and then one has to refer to
context to know what is meant.  It is never wrong to ask!