# Thread: definition of topological space, problem with understanding

1. ## definition of topological space, problem with understanding

Hello! I am trying to understand the definition of topological space but I have some difficulties with it

Here it comes

Let $\displaystyle X$ be anyset and $\displaystyle T={U_{i}|i\in I}$ denote a certain collection of subsets of $\displaystyle X$. The pair $\displaystyle (X,T)$ is called a topological space if $\displaystyle T$ atisfies the following requirements:
(i) $\displaystyle \emptyset, X\in T$
(ii) if $\displaystyle T$ denotes any (maybe infinite) subcollection of $\displaystyle I$ , the family$\displaystyle {U_{j}|j\in J$ satifies $\displaystyle \cup_{j\in J}U_{j}\in T$.
(iii) if $\displaystyle K$ is any finite subcollection of $\displaystyle I$ the family $\displaystyle {U_{k}|k\in K$ satisfies $\displaystyle \cap_{k\in K}U_{K}\in T$

well I just do not get it.
1)Does it mean that [tex] T is some collection of susbsets $\displaystyle U_{i}$ so that the union of this subsets belongs to T (covers it?)

2)I understand that X has to belong to T but why even the empty set has to be there?

3) could someone give me descriptive interpretation of the conditions (ii) and (iii)

Can we just say that the first one requires the empty set and our set X to be in T
second one says that the union of any collections of sets in T is also in T
and the last one says that the intersection of any pair of sets in T is also in T?

But i still do not know why the empty set with X belongs to T???

2. Wikipedia is your friend! Seriously, though, they do a great job of describing topological spaces in that article.

3. Originally Posted by SlipEternal
Wikipedia is your friend! Seriously, though, they do a great job of describing topological spaces in that article.
actually I am reading it now, and there is nothing about why the empty set has to be there, is it just there and I take as it it, it just has to be there. There are also 6 interesting examples , 2 of them are not topologies, but I still do not see why they should not be topologies.

Wikipedia though has much easier definition than the one I have in my book.

4. Topological spaces are used in definitions for many things. Check out some of the things topological spaces are used to define and the need for the empty set should become apparent.

Essentially, pick a nonempty subset of $\displaystyle X$ contained in $\displaystyle T$. You can begin breaking down that subset by taking its intersection with any subset of X (in $\displaystyle T$ or not) and the intersection is still in $\displaystyle T$

5. Originally Posted by rayman
there is nothing about why the empty set has to be there
First a bit of terminology. The sets in $\displaystyle \mathcal{T}$ are called basic open sets.
We can have disjoint basic open sets.
Their intersection is $\displaystyle \emptyset$.
But requirement (iii), finite intersections are in $\displaystyle \mathcal{T}$.
So $\displaystyle \emptyset\in\mathcal{T}$ must be true.

6. Thank you , I think I am beginning to understand it.
Everyone who like me study topology on their own and struggles a bit with understanding the basis of topology check this out
YouTube - &#x202a;Topology #10 Topology Examples&#x202c;&rlm;

great series of short presentations where the author explains some of concepts in a very nice way