Topology/Continuity and Homeomorphisms

Continuity
Continuity is the central concept of topology. Essentially, topological spaces have the minimum necessary structure to allow a definition of continuity. Continuity in almost any other context can be reduced to this definition by an appropriate choice of topology.

Definition
Let $$X,Y$$ be topological spaces.

A function $$f:X\to Y$$ is continuous at $$x\in X$$ if and only if for all open neighborhoods $$B$$ of $$f(x)$$, there is a neighborhood $$A$$ of $$x$$ such that $$A\subseteq f^{-1}(B)$$. A function $$f:X\to Y$$ is continuous in a set $$S$$ if and only if it is continuous at all points in $$S$$.

The function $$f:X\to Y$$ is said to be continuous over $$X$$ if and only if it is continuous at all points in its domain.

$$f:X\to Y$$ is continuous if and only if for all open sets $$B$$ in $$Y$$, its inverse $$f^{-1}(B)$$ is also an open set.

Proof:

($$\Rightarrow$$)

The function $$f:X\to Y$$ is continuous. Let $$B$$ be an open set in $$Y$$. Because it is continuous, for all $$x$$ in $$f^{-1}(B)$$, there is a neighborhood $$x\in A\subseteq f^{-1}(B)$$, since B is an open neighborhood of f(x). That implies that $$f^{-1}(B)$$ is open.

($$\Leftarrow$$)

The inverse image of any open set under a function $$f$$ in $$Y$$ is also open in $$X$$. Let $$x$$ be any element of $$X$$. Then the inverse image of any neighborhood $$B$$ of $$f(x)$$, $$f^{-1}(B)$$, would also be open. Thus, there is an open neighborhood $$A$$ of $$x$$ contained in $$f^{-1}(B)$$. Thus, the function is continuous.

If two functions are continuous, then their composite function is continuous. This is because if $$f$$ and $$g$$ have inverses which carry open sets to open sets, then the inverse $$g^{-1}(f^{-1}(x))$$ would also carry open sets to open sets.

Examples

 * Let $$X$$ have the discrete topology. Then the map $$f:X \rightarrow Y$$ is continuous for any topology on $$Y$$.
 * Let $$X$$ have the trivial topology. Then a constant map $$g:X \rightarrow Y$$ is continuous for any topology on $$Y$$.

Homeomorphism
When a homeomorphism exists between two topological spaces, then they are "essentially the same", topologically speaking.

Definition
Let $$X,Y$$ be topological spaces

A function $$f:X\to Y$$is said to be a homeomorphism if and only if

(i) $$f$$ is a bijection

(ii) $$f$$ is continuous over $$X$$

(iii)$$f^{-1}$$ is continuous over $$Y$$

If a homeomorphism exists between two spaces, the spaces are said to be homeomorphic

If a property of a space $$X$$ applies to all homeomorphic spaces to $$X$$, it is called a topological property.

Exercises
(ii)Determine whether this $$f$$ is a homeomorphism.
 * 1) Prove that the open interval $$(a,b)$$ is homeomorphic to $$\mathbb{R}$$.
 * 2) Establish the fact that a Homeomorphism is an equivalence relation over topological spaces.
 * 3) (i)Construct a bijection $$f:[0,1]\to [0,1]^2$$