Bijections and Visual Representations

Question

Throughout math we learn about the cardinality of sets. We learn that the existence of a bijection between two sets imply that the cardinality of these two sets are equal. For each set there is similar to a picture we can draw to represent it. For example, when drawing the set $\mathbb{R}$ we often draw a straight line.

That brings me to my question:

Let us say that we have a bijection $f: \mathbb{R} \rightarrow A$ for some set $A$. Does this mean we can represent $A$ as a real line as well? It seems that anything that holds for $\mathbb{R}$ should hold for $A$, however that might not be the case.

If this is not true for a bijection, then is it true for two groups that are isomorphic? Or maybe there is another term that I am unaware of that denotes we can represent two sets as the same picture?

Answer 1

When we depict $\mathbb{R}$ as a straight line, we're usually not just thinking of it as a set, we're thinking of it as a space: it comes equipped with a huge amount of extra structure, such a notion of distance, arithmetic operations, and so on.

Bijections don't preserve all this structure, they simply preserve the number of elements. For example, there is a bijection $\mathbb{R} \to \mathcal{P}(\mathbb{N})$, but $\mathcal{P}(\mathbb{N})$ doesn't retain the same structure as $\mathbb{R}$. You could, if you wanted to, depict $\mathcal{P}(\mathbb{N})$ as a line, but the line wouldn't have any meaning in the usual sense.

However, there are generalisations of the notion of a bijection that might get at what you want. For example:

• If $X$ is a metric space (i.e. a set equipped with a notion of distance), then a bijection $\mathbb{R} \to X$ that preserves the distance is called a (global) isometry;
• If $X$ is a topological space (i.e. a set equipped with a notion of nearness, which is a bit weaker than distance), then a bijection $\mathbb{R} \to X$ that preserves this structure is called a homeomorphism;
• If $X$ is a ring or a field (i.e. a set equipped with a notion of arithmetic), then a bijection $\mathbb{R} \to X$ that preserves all the arithmetic operations is called a ring (or field) homomorphism.

In the first two cases (where we consider $\mathbb{R}$ as a certain kind of space), the set $X$ will in some sense look like a line.