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Let $X$ be a topological space with a fixed topology $\mathscr{T}$. We know that the following are equivalent for all $U \subseteq X$.

  1. $U \in \mathscr{T}$.
  2. For all $x \in U$ there is $U_{x} \in \mathscr{T}$ such that $x \in U_{x} \subseteq U$.

My question is, is it okay to define topology as follows?

Definition. Let $X$ be a set. A subset of power set $\mathscr{T} \subseteq \mathcal{P}(X)$ is called a topology on $X$ if for every $U \in \mathscr{T}$ the following is true: for all $x \in U$, there exists $U_{x} \in \mathscr{T}$ such that $x \in U_{x} \subseteq U$.

If this is okay, I don't know what made most of people introduce topology with three axioms. Is it because this definition is self-referencing? Before I sat in my introductiory topolgy course, I always thought that this was going to be introduced in the beginning of the class rather than more axiomatic definition.

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    What is $U$ in your definition?2012-10-18
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    Your definition does not require the union of open sets to be open. Isn't this something desirable?2012-10-18
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    The role of axioms in mathematics is to give a rigorous framework to describe properties of objects. Some long time ago, people figured what are the properties we expect open sets to have, and ended up with a very concise way of saying that. Not everything needs to be recursive, and not everything *can* be recursive.2012-10-18
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    @Chris Thanks, I meant to say for all $U \in\mathscr{T}$. Let me fix it right away.2012-10-18
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    @AndresCaicedo - the definition had a typo. Sorry for the confusion.2012-10-18
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    Yes, that was clearly your intent. Still, unions of open sets need not be open. And it could be that only the empty set is open. Do you want either possibility to hold?2012-10-18
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    @AsafKaragila - Thanks and I started to see some of those recently. I just thought that this recursive definition seems to be more intuitive just because of some calculus knowledge. However, I don't know what made many people say that axiomatic definitions are more powerful than the others. I do agree that they are easy to work with but are there other reasons that it gives more than other definitions?2012-10-18
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    @AndresCaicedo the empty set is clearly open (according to the recursive definition), but I don't see other claims that you made. Can you explain more? That's basically why I answered this question.2012-10-18
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    Never mind, I see now. Thanks all!2012-10-18

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No. For one, any covering of $X$ would be a topology. Alternatively, the set of singletons (not the discrete topology) would be a topology on every space by your logic.

You need to be able to take unions and stay inside the topology to do anything remotely interesting.

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    That makes sense. So recursive definition has more sets in $\mathscr{T}$. What a stupid question. I totally forgot the fact that we are already given more information that $\mathscr{T}$ is a topology in the very first logical equivalence. Thanks!2012-10-18
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This definition only guarantee you that all $U \subset \mathscr{T}$ are open, this is because in the definition of topology is important:

  1. Both the empty set and $X$ are elements of $\mathscr{T}$
  2. Any union of elements of $\mathscr T$ is an element of $\mathscr{T}$
  3. Any intersection of finitely many elements of $\mathscr{T}$ is an element of $\mathscr{T}$

This is no recursively. And the first part tell you that if you have any set, maybe a closed set, you can find a open set of any point of that set that belong to the fixed topology.

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    I don't think this answer is relevant to what I asked. See anon's answer.2012-10-18
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    "And the first part tell you that if you have any set, maybe a closed set, you can find a open set of any point of that set that belong to the fixed topology." - What do you mean?2012-10-19
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    If U is open then $U \in \mathscr{T}$, but if U is closed for all $x \in U$, you can find $U_{x} \in \mathscr{T}$ which is open by definition.2012-10-19
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    Note that I did not use the definition of the word "closed" as one should not use it as we are discussing the definition of the word "open."2012-10-19
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    I used the set closed as a example and it's evident the definition.2012-10-19
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    It is not evident when you don't know what you mean what the word "closed" means. Note that the discussion is about defining the word "open" and usually we say a set $E$ is closed if $X \setminus E$ is open when $X$ is understood as the topological space that we are working in.2012-10-19
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    Thus your example does not make much sense to me, unless you use a different definition of closedness, which is not very efficient for the sake of this discussion.2012-10-19
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    If $U \notin \mathscr{T}$ then U is closed or neither closed nor open, but in the example I assume the set closed... And $\mathscr{T}$ is fixed.2012-10-19