1
vote
0answers
33 views

Can one talk about equivalences classes in a class(which might not be a set). [duplicate]

equivalence relation on a set breaks it into disjoint sets. does 'disjointness' make sense in classes. specifically, given a category, isomorphism classes of objects makes sense or not. can it be so ...
3
votes
1answer
47 views

What is the common preimage (in $Z$) and the equivalence relation for Pushouts

Here it says: Suppose that $X$, $Y$, and $Z$ as above are sets, and that $f : Z → X$ and $g : Z → Y$ are set functions. The pushout of $f$ and $g$ is the disjoint union of $X$ and $Y$, where elements ...
8
votes
3answers
172 views

Can we take images of equivalence relations?

Given a function $f : X \rightarrow Y$, it is well-known that we can take the image under $f$ of any subset $A \subseteq X$, and we can take the preimage under $f$ of any subset $A \subseteq Y$. This ...
1
vote
1answer
72 views

Understanding Pushouts in Top.

The Pushout of $X \leftarrow Z\rightarrow Y$ with $f:Z\rightarrow X$ and $g:Z\rightarrow Y$ in $\mathbf{Top}$ exists and is given by $X\coprod Y/\sim,$ where "$\sim$ is the equivalence relation ...
4
votes
2answers
93 views

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

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 ...
2
votes
2answers
437 views

Distinguishing equality and isomorphism as relations

Is this relational characterization of equality in Wikipedia accepted? The identity relation is the archetype of the more general concept of an equivalence relation on a set: those binary ...