From a topological perspective a deformation or manipulation of a 1 ft rubber into various shapes is absolutely still the original 1 ft rubber balloon.

The Core Principle: Homeomorphism

Topology is often called "rubber sheet geometry." It is the branch of mathematics that studies the properties of objects that remain unchanged when they are stretched, twisted, bent, or otherwise deformed, as long as you don't tear, cut, or glue parts together.

The key concept here is called a homeomorphism. Two objects are considered topologically equivalent (or "the same") if one can be deformed into the other without cutting or gluing.

Your scenario is a perfect, real-world example of this principle.

Applying the Principle to Your Balloon

Let's trace the transformations:

You start with a sealed, 1 ft rubber balloon. Topologically, this is a sphere (S²). The air inside and the thickness of the rubber are irrelevant; we only care about the surface.

You manipulate it into a dog, a swan, or a sword. To make the dog, you stretch and pinch parts to form legs, a tail, and a head. You are not cutting holes for the mouth or eyes, and you're not poking a separate tail and gluing it on. You are just deforming the continuous surface. To make the sword, you stretch the balloon into a long, thin cylinder for the blade and pinch a handle. Again, no cuts or joins.

At every stage, the object remains a single, continuous, sealed surface with no holes.

Because you can continuously deform the sphere (the balloon) into the dog-shape and back again without any cutting or gluing, the two shapes are homeomorphic. They are, topologically, identical.

What Would Change Its Topology?

If you were to perform an action that changes the fundamental topology, it would no longer be the same.

For example:

If you popped the balloon, you would have a torn, flat piece of rubber. A disk is not topologically the same as a sphere.

If you untied the neck and opened it up, you would have a different shape (perhaps a disk if you flattened it). A sphere with a hole is not a sphere.

If you somehow tied the balloon into a torus (donut shape), that would require joining two parts of the surface together in a way that creates a permanent hole. This is not possible with a simple balloon without cutting and re-gluing, so it doesn't apply to your scenario.

A Helpful Analogy: The "Sock Monkey"

Imagine a knitted sock. It is topologically a distorted cylinder (or a sphere if you consider the closed toe).

You can stuff it and sew it into a sock monkey with long arms, legs, and a tail. As long as you are only sewing parts together (which is like "gluing" in topology, and is allowed as a transformation for homeomorphism as long as it's reversible by cutting), the surface of the sock monkey is still a single, continuous surface. It is still topologically equivalent to the original sock.

Conclusion

So, while the balloon's geometry (its exact shape, size, curvature) changes dramatically as you make your animals and objects, its topology remains constant. It is a closed, continuous surface with no holes—a sphere.

Therefore, from the perspective of topological maths, you are merely looking at the original 1 ft rubber balloon in a different, fantastically creative configuration.

In short: The shape is new, but the underlying topological "soul" of the object remains the same.