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Open Sets: Topology Without Distance

A metric gives you distance, and distance gives you open balls, and open balls give you open sets. But the punchline of point-set topology is that you can throw the first two away and keep the third. We axiomatize the open sets directly and see that this short list of rules already carries a surprising amount of analysis.

From open balls to open sets

In a metric space you already know what an open ball is: B(x, r) is the set of points within distance r of x. A set U is called open when every point of U sits inside some open ball that is still entirely contained in U — informally, every point has a little breathing room. Now watch which facts about open sets we actually used in proofs. We used that the whole space and the empty set are open; that the union of any collection of open sets is open; and that the intersection of two open sets is open. Almost everything else followed from just these three.

That observation is the whole idea of topology. Instead of starting from a distance and deriving the open sets, we start from the open sets themselves, declaring by fiat which subsets count as open, subject only to those three rules. Distance never appears.

The open-set axioms

A topology on a set X is a collection τ of subsets of X — the ones we agree to call open — satisfying exactly the three axioms below. The pair (X, τ) is a topological space. Note the asymmetry: arbitrarily many unions are allowed, but only finitely many intersections.

  1. (T1) The empty set ∅ and the whole space X are both open.
  2. (T2) Any union of open sets is open — even an infinite or uncountable union.
  3. (T3) The intersection of finitely many open sets is open.
Claim: in any topological space, a finite intersection of open sets is open.

Proof by induction on the number of sets k.

Base case k = 2:  U1 and U2 open => U1 ∩ U2 open  (this is axiom T3).

Inductive step: suppose any intersection of k open sets is open.
Let U1, ..., U_{k+1} be open. Write
    U1 ∩ ... ∩ U_{k+1} = (U1 ∩ ... ∩ Uk) ∩ U_{k+1}.
By the inductive hypothesis V := U1 ∩ ... ∩ Uk is open.
Then V ∩ U_{k+1} is an intersection of TWO open sets, hence open by T3.

So the statement holds for k+1, and by induction for every finite k.   ∎

Note: the proof breaks for infinite intersections — there is no "last" set
to peel off, and the induction never reaches it. T3 really is finite-only.
Finite intersections of open sets are open — a clean induction, and a reminder of where it stops.

Neighbourhoods and the simplest examples

A neighbourhood of a point x is any open set containing x (some books say any set that contains such an open set). Neighbourhoods replace the metric notion of being “near” x: a property holds near x if it holds throughout some neighbourhood. Two extreme topologies bracket every other one. The discrete topology makes every subset open — points are maximally separated. The indiscrete (trivial) topology has only ∅ and X open — points are inseparable. The interesting topologies live in between.