1. ## basic topology trouble

I am reading 'Basic topology- M. A. Armstrong' and in Chapter 2 my book defines:

Open set: Let $X$ be a topological space then $O \subset X$ is an open set if its a neighborhood of each of its points.
Neighborhood: $N \subset X$ is a neighborhood of $x \in X$ if we can find an open set $O$ such that $x \in O \subset N.$

To define open sets we needed neighborhoods and to define neighborhoods we needed open sets. This seems ambiguous to me. Help needed on this.
Also if you know a good book on Topology for Beginners then please tell me. I have read 'A first course in Real Analysis- Sterling K. Berberian'.

2. ## Re: basic topology trouble

That does seem a bit circular. Any chance you could scan in and post the pages that have those two definitions?

Off-hand, I would say that, typically, open sets are defined as the members of a topological space as follows: a topological space is a set, X, together with a collection, T, of subsets of X, called "open" sets, which satisfy the following rules:

T1. The set X itself is "open"
T2. The empty set is "open"
T3. Arbitrary unions of "open" sets are "open"
T4. Finite intersections of "open" sets are "open".

- Martin D. Crossley, Essential Topology, p. 15.

You can define open sets in $\mathbb{R}$ and higher-dimensional spaces (really, any metric space) by looking at open balls, which is a more fundamental concept than open sets. In that setting, I would define a neighborhood, N, of x as a set such that there exists an open ball containing x that is itself contained in N. The concept of open balls does not depend on open sets, but it does need a metric.

The Crossley book, incidentally, is one I would recommend. Explanations seem very straight-forward to me.

3. ## Re: basic topology trouble

Originally Posted by abhishekkgp
Open set: Let $X$ be a topological space then $O \subset X$ is an open set if its a neighborhood of each of its points.
It can't be the definition of an open set : the term "topological space" implies the notion of open sets ! Indeed, if (X,T) is a topological space, as Ackbeet said, an open set is the name of the element of T.

So your quote is just a property of an open set, not a definition.

4. ## Re: basic topology trouble

Originally Posted by Ackbeet
That does seem a bit circular. Any chance you could scan in and post the pages that have those two definitions?

Off-hand, I would say that, typically, open sets are defined as the members of a topological space as follows: a topological space is a set, X, together with a collection, T, of subsets of X, called "open" sets, which satisfy the following rules:

T1. The set X itself is "open"
T2. The empty set is "open"
T3. Arbitrary unions of "open" sets are "open"
T4. Finite intersections of "open" sets are "open".

- Martin D. Crossley, Essential Topology, p. 15.

You can define open sets in $\mathbb{R}$ and higher-dimensional spaces (really, any metric space) by looking at open balls, which is a more fundamental concept that open sets. In that setting, I would define a neighborhood, N, of x as a set such that there exists an open ball containing x that is itself contained in N. The concept of open balls does not depend on open sets, but it does need a metric.

The Crossley book, incidentally, is one I would recommend. Explanations seem very straight-forward to me.
thank you Ackbeet. I will check out the book you recommended.

5. ## Re: basic topology trouble

Originally Posted by abhishekkgp
thank you Ackbeet. I will check out the book you recommended.
You're welcome!

6. ## Re: basic topology trouble

The standard book recommendation (and it's the standard for good reason) is Topology by Munkres.