In topology $X$ is also $Y$ means homeomorphic?

Question

E.g. $\mathbb{R}P^n$ is also the quotient space $S^n / (v \sim -v)$. And when is it safe to refer to a space as one of it's homeomorphic spaces and perform further deductions from that homeomorphic space?

Answer 1

All of the topological information of a topological space is contained in its open sets, or equivalently, its closed sets. What I mean by this is that any topological property is completely determined by the topology, ie, the open sets.

What is a topological property? Well, at the risk of being tautological, it is exactly a property that is defined using only the notion of closed and open sets. Since two homeomorphic spaces have exactly the same open and closed sets, they will also contain exactly the same topological properties.

Let's give a few examples of topological properties.

Examples:

1. Compact
2. Connected
3. Hausdorff
4. Separable
5. Normal
6. Discrete
7. Regular

Now, if the topological space has more structure, like a metric, then properties that depend on the metric, like completeness, are not topological properties, and therefore might not be preserved by a homeomorphism. Notice that $\mathbb{R}$ is homeomorphic to $(0,1)$, but the open interval is not complete, as $\mathbb{R}$ is.

So if you remain totally in the realm of topology, there is no danger in studying a space $X$ to learn about a space $Y$ if $X\cong Y$.

Answer 2

Plain isomorphism can be good enough if you study one object alone but this can fail in families (such as nontrivial bundles). Natural isomorphism is almost always enough to treat isomorphic objects as equal.

This is assuming you are interested in isomorphism-invariant properties, and the notion of isomorphism is taken in whatever category accomodates the constructions to be done on $X$ and its isomorphs.