2

I was trying to learn disjoint union topology and used the following blog :

https://drexel28.wordpress.com/2010/04/02/disjoint-union-topology/

The second theorem about disjoint topology says that if {$\ X_j$} where $\ j$ $\ \varepsilon $$\ J$ be a class of topological spaces such that $\ X_j$$\ \approx$ $\ \mathcal X $ with $\ \psi_j$:$\ X_j$$\ \to$ $\ \mathcal X $ the homeomorphism.Then $\ \coprod$$\ X_j$ has the D.U.T and $\ \tau$ the discrete.

What does $\ X_j$$\ \approx$$\ \mathcal X$ mean here ? Also I would like to know the mathematical definition of the word "class " and the places where we should use it instead of sets .

marshal craft
  • 700
jedna dve
  • 181
  • Classes are very similar to sets. However with sets there came to be a problem when they tried to aximatize set theory. The problem is if you have the set of all sets. Then that set is an element of another set causing a contradiction. The resolution to this was to envision an object which fixes this, called a class. There is the class of all sets. But by the definition of a class, a class need not be a set and doesn't necessarily have a super set. Thus there is no contradiction. In topology it is likely used instead of a set because topology deals closely with formal sets and set theory. – marshal craft May 21 '16 at 06:17
  • I recommend for now, in all intents and purpose to just view the class as another set. You will distinguish the two when the need arises and it will then make sense. – marshal craft May 21 '16 at 06:20
  • @marshalcraft : Thanks a lot . But how does the envisioning of an object like class help ? – jedna dve May 21 '16 at 07:42
  • When considering statements which involve the intuitive "collection" of all sets for example; the exact cases which make the distinguishable necessary (and thus the creation of them in the first place). Perhaps beyond the scope of, and independent of your primary question. – marshal craft May 21 '16 at 07:49

1 Answers1

1

Working from the linked blog, $X_j \approx \mathcal X$ means that $X_j$ is homeomorphic to $\mathcal X$. This makes sense, as he then immediately gives a name to a homeomorphism.

Classes are confusing. But there is a lot written about them already on MSE. I direct you to the question Difference between a class and a set as a good jumping point here. Following up on the linked questions will lead you to lots of good exposition.

Community
  • 1
davidlowryduda
  • 91,687