Topology/Continuity and Homeomorphisms

Continuity

Continuity is the central concept of topology. Essentially, topological spaces have the minimum necesary 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

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

The function f:X→Y itself is continuous if and only if it is continuous at all points in its domain.

The function f:X→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:
→
The function f:X→Y is continuous. Let B be a open set in Y. Because it is continuous, for all x in f^-1(B), there is a neighborhood A⊆f^-1(B). That implies that it is continuous.
←
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 $f:X\to Y$ is continuous for any topology on $Y$.
• Let $X$ have the trivial topology. Then a constant map $g:X\to Y$ 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

if x1 x2,.....,xn subset of x consider :- x1* $f:X\to Y$ is a homeomorphism if it is continuous, and has a continuous inverse f^-1 such that f^-1(f(x)) is the identity function for X and such that f(f^-1(x)) is the identity function for Y.

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

Consider any homeomorphism f. For any x in X, x=f^-1(f(x)) and for any y in Y, y=f(f^-1(y)), thus proving that they are functions, and thus they are one-to-one. This also implies that their domain is the whole space, implying that it is onto. Thus, they are bijections.

Homeomorphism is an equivalence relation, because all spaces are homeomorphic to themselves under the identity relation, the inverse function of a homeomorphism is also a homeomorphism, and the composition of two homeomorphisms is also a homeomorphism (because the composition of two continuous functions is also continuous).

Counterexample

Note that a map may be bijective and continuous, but not a homeomorphism. Consider the bijective map $f:[0,1)\to S^1$, where f(x)=e^2πix mapping the points in the domain onto the unit circle in the plane. This is not a homeomorphism, because there exist open sets in the domain that are not open in $S^1$, like the set [0,½).

Exercises

1. Prove that the open interval (a,b) is homeomorphic to R.