Thread: basic topology trouble

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

Open set: Let be a topological space then is an open set if its a neighborhood of each of its points.
Neighborhood: is a neighborhood of if we can find an open set such that

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'.

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 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.

Originally Posted by abhishekkgp

Open set: Let be a topological space then 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.

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 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.

Originally Posted by abhishekkgp

thank you Ackbeet. I will check out the book you recommended.

You're welcome!

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