3 added 47 characters in body

This is not essentially different from what everyone else has said, but for some reason I feel like saying it anyway.

The number of times that a given topological space appears in the category of topological spaces is exactly once. That's the definition of the class of objects of Top.

As for isomorphism classes: there is exactly one object in Top which is isomorphic to the empty set $\varnothing$ with its unique topology: namely $\varnothing$. For any nonempty topological space $X$, the subclass of Top consisting of spaces $X'$ which are homeomorphic to $X$ forms a proper class: i.e., they are more numerous than any set. This is true because the class of all sets is a proper class, and if $(X,\tau)$ is any nonempty topological space and $S$ is any set, then $X \times \{S\}$ is a set which is different from $X$ but in bijection with it: $x \mapsto (x,S)$, and the image of $\tau$ under this bijection gives a topological space which is homeomorphic to $(X,\tau)$ but with a different underlying set.

So, except for one, the isomorphism homeomorphism classes in Top are "unsetically" huge. Nevertheless, the modern perspective is that this is fine (or, perhaps, harmlessly irrelevant) whereas it is a bad idea to pass to work with the homeomorphism classesclass of the space instead of the space itself. One way to think about this is that a given topological space $X$ is a relatively simple object, but the class of all topological spaces homeomorphic to $X$ is a ridiculously complicated object. (This has not always been the received wisdom: notably Bertrand Russell's definition of the number $2$ was the class of all sets which can be put in bijection with, say, $\{\emptyset, \{ \emptyset\} \}$.) From a less philosophical perspective, one wants to speak about sets of maps between two topological spaces $X$ and $Y$, and if $X$ and $Y$ are only well-defined up to a homeomorphism, these sets are themselves not so well defined. As soon as one studies commutative diagrams of morphisms of objects in a category (or more generally, functorial constructions), one recognizes that the concept of "topological space up to a homeomorphism" is a painfully awkward one.

It is also simply against the spirit of category theory to pass to isomorphism classes: many would say that this overemphasizes the somewhat quaintly philosophical notion of equality of objects. Instead of saying "the topological space $X$ is equal to the topological space $Y$", many mathematicians now think it is both simpler and more useful to say "$\Phi: X \rightarrow Y$ is a homeomorphism". For more on this point, I highly recommend Barry Mazur's article When is one thing equal to some other thing?

2 added 83 characters in body

This is not essentially different from what everyone else has said, but for some reason I feel like saying it anyway.

The number of times that a given topological space appears in the category of topological spaces is exactly once. That's the definition of the class of objects of Top.

As for isomorphism classes: there is exactly one object in Top which is isomorphic to the empty set $\varnothing$ with its unique topology: namely $\varnothing$. For any nonempty topological space $X$, the subclass of Top consisting of spaces $X'$ which are homeomorphic to $X$ forms a proper class: i.e., they are more numerous than any set. This is true because the class of all sets is a proper class, and if $(X,\tau)$ is any nonempty topological space and $S$ is any set, then $X \times \{S\}$ is a set which is different from $X$ but in bijection with it: $x \mapsto (x,S)$, and the image of $\tau$ under this bijection gives a topological space which is homeomorphic to $(X,\tau)$ but with a different underlying set.

So, except for one, the isomorphism classes in Top are "unsetically" huge. Nevertheless, the modern perspective is that this is fine (or, perhaps, harmlessly irrelevant) whereas it is a bad idea to pass to homeomorphism classes. One way to think about this is that a given topological space $X$ is a relatively simple object, but the class of all topological spaces homeomorphic to $X$ is a ridiculously complicated object. (This has not always been the received wisdom: notably Bertrand Russell's definition of the number $2$ was the class of all sets which can be put in bijection with, say, $\{\emptyset, \{ \emptyset\} \}$.) From a less philosophical perspective, one wants to speak about sets of maps between two topological spaces $X$ and $Y$, and if $X$ and $Y$ are only well-defined up to a homeomorphism, these sets are themselves not so well defined. As soon as one studies commutative diagrams of morphisms of objects in a category (or more generally, functorial constructions), one recognizes that the concept of "topological space up to a homeomorphism" is a painfully awkward one.

It is also simply against the spirit of category theory to pass to isomorphism classes: many would say that this overemphasizes the somewhat quaintly philosophical notion of equality of objects. Instead of saying "the topological space $X$ is equal to the topological space $Y$", many mathematicians now think it is both simpler and more useful to say "$\Phi: X \rightarrow Y$ " is a homeomorphismhomeomorphism". For more on this point, I highly recommend Barry Mazur's article When is one thing equal to anothersome other thing?

1

This is not essentially different from what everyone else has said, but for some reason I feel like saying it anyway.

The number of times that a given topological space appears in the category of topological spaces is exactly once. That's the definition of the class of objects of Top.

As for isomorphism classes: there is exactly one object in Top which is isomorphic to the empty set $\varnothing$ with its unique topology: namely $\varnothing$. For any nonempty topological space $X$, the subclass of Top consisting of spaces $X'$ which are homeomorphic to $X$ forms a proper class: i.e., they are more numerous than any set. This is true because the class of all sets is a proper class, and if $(X,\tau)$ is any nonempty topological space and $S$ is any set, then $X \times \{S\}$ is a set which is different from $X$ but in bijection with it: $x \mapsto (x,S)$, and the image of $\tau$ under this bijection gives a topological space which is homeomorphic to $(X,\tau)$ but with a different underlying set.

So, except for one, the isomorphism classes in Top are "unsetically" huge. Nevertheless, the modern perspective is that this is fine (or, perhaps, harmlessly irrelevant) whereas it is a bad idea to pass to homeomorphism classes. One way to think about this is that a given topological space $X$ is a relatively simple object, but the class of all topological spaces homeomorphic to $X$ is a ridiculously complicated object. (This has not always been the received wisdom: notably Bertrand Russell's definition of the number $2$ was the class of all sets which can be put in bijection with, say, $\{\emptyset, \{ \emptyset\} \}$.) From a less philosophical perspective, one wants to speak about sets of maps between two topological spaces $X$ and $Y$, and if $X$ and $Y$ are only well-defined up to a homeomorphism, these sets are themselves not so well defined. As soon as one studies commutative diagrams of morphisms of objects in a category (or more generally, functorial constructions), one recognizes that the concept of "topological space up to a homeomorphism" is a painfully awkward one.

It is also simply against the spirit of category theory to pass to isomorphism classes: many would say that this overemphasizes the somewhat quaintly philosophical notion of equality of objects. Instead of saying "the topological space $X$ is equal to the topological space $Y$", many mathematicians now think it is both simpler and more useful to say "$\Phi: X \rightarrow Y$" is a homeomorphism. For more on this point, I highly recommend Barry Mazur's article When is one thing equal to another?