Homeomorphisms and topological invariants

Try

The open interval (0,1) and the whole real line have different lengths and boundedness, but they are topologically equivalent. The map h(x)=log(x/(1−x)) stretches the interval continuously onto R, with continuous inverse exp(y)/(1+exp(y)).

Topology ignores numerical distances unless they affect open sets. Boundedness in a chosen metric is therefore not a general topological invariant. The example is a useful warning against deciding equivalence from the size or appearance of a drawing.

Predict

Predict whether every continuous bijection is enough to establish equivalence. Consider the identity from discrete R to usual R. It is continuous and bijective, but its inverse is not continuous because a discrete singleton has a nonopen usual preimage.

The function formula and the set-theoretic inverse exist in both directions. What fails is the inverse’s compatibility with the specified open sets.

Experiment

Compare three requirements.

Property	What must be checked
Bijection	Exactly one preimage for every target point
Continuous bijection	In addition, target opens pull back to source opens
Homeomorphism	The inverse also pulls opens back to opens

For a bijection, inverse continuity is equivalent to the original map sending opens to opens. This is a special relationship for a bijection, not the definition of continuity for arbitrary maps.

Understand

A homeomorphism h:X→Y is a bijection with both h and h⁻¹ continuous. It transports the full open-set structure: U⊆X is open exactly when h(U) is open in Y. The forward direction uses inverse continuity; the reverse uses continuity of h.

For a continuous bijection, being an open map is equivalent to being a homeomorphism. Indeed, the preimage of U under the inverse is h(U). Thus the open-map condition is exactly the missing inverse-continuity condition. Replacing opens by closed sets gives the analogous closed-map test.

A topological invariant is a property preserved by homeomorphisms. Discreteness provides a first proof example. If X is discrete, every subset of Y is h(A) for some open A⊆X; openness of h makes every subset of Y open. Thus Y is discrete. This rules out a homeomorphism between usual R and discrete R without analyzing every possible bijection.

Later, compactness, connectedness, separation properties and countability conditions will supply more informative invariants. A failed proposed formula does not by itself prove two spaces are nonhomeomorphic; an invariant rules out all possible homeomorphisms at once.

For the interval-to-line example, explicitly solving the formula gives the inverse. Its continuity and values in (0,1) prove the result more directly than separate endpoint and monotonicity arguments, although those provide useful checks.

Apply

Verify h and its inverse at x=1/2 and y=0, then prove the inverse identities algebraically. Explain why endpoint behavior tending to infinity is compatible with continuity: the endpoints are not points of the source space.

Repair “A homeomorphism preserves straight lines and angles.” Those are geometric features of a coordinate or metric structure. A homeomorphism preserves topology; stronger notions such as linear, isometric or conformal maps preserve additional structure under additional hypotheses.

Continue

Products and quotients will construct new spaces whose equivalence can be tested by these criteria. Optional depth: prove that compositions and inverses of homeomorphisms are homeomorphisms. This makes topological equivalence an equivalence relation, while leaving the task of finding or excluding a specific equivalence mathematically substantive.