Basic Topology 2 | Topological Spaces

TL;DR

A topology on X is a collection T of subsets of X that includes ∅ and X, is closed under finite intersections, and is closed under arbitrary unions.

Briefing Cornell Notes

Briefing

Topology starts with a simple move: take an arbitrary set X and decide which subsets of X should count as “open.” Those chosen subsets form a topology, a collection T ⊆ P(X) (the power set of X) that must satisfy three rules: the empty set and X itself are always in T; the intersection of any two open sets is also open; and the union of any collection of open sets is open, even when the collection is infinite or uncountable. With these closure properties, “open” becomes a structural notion rather than something inherited from geometry.

Once a topology T is fixed, the pair (X, T) is a topological space. The same underlying set X can support many different topologies, and changing the topology can radically change what it means for points to be “near,” which sets to be “large,” and how limits behave. In other words, topology is less about the points themselves and more about the rules for which subsets are considered open.

The course then connects this abstract framework to the familiar setting of metric spaces. On R^n, the standard Euclidean norm ||x|| (described as measuring vector length) induces a metric by turning distances into norms of difference vectors. Any metric, in turn, generates a topology: a set A is open (with respect to the metric) exactly when every point in A sits inside some ε-ball that stays entirely within A. Those ε-balls are the geometric engine behind openness.

A key subtlety follows: the shape of ε-balls depends on the chosen norm. Switching from one norm to another can make the balls look different—squares versus circles, for instance—but the resulting notion of openness on R^n does not change. So the “standard topology” on R^n is robust: it can be obtained from any norm on R^n, even though the geometry of balls varies.

Finally, the discussion emphasizes that topology need not come from a norm or metric at all. One extreme example is the indiscrete topology, where only ∅ and X are open. In that topology, every point is effectively as close as possible to every other point, making convergence and “neighborhood” ideas behave in a very degenerate way. The indiscrete topology also illustrates that not every topology is metric-induced; some topologies cannot be produced by any metric. The groundwork is laid for the next steps—generalizing closed sets and compactness beyond the standard metric setting—while noting that metric-induced topologies that differ from the standard one will be addressed later.

Cornell Notes

A topology on a set X is a collection T of subsets of X (so T ⊆ P(X)) that includes ∅ and X, is closed under finite intersections, and is closed under arbitrary unions. Declaring which subsets are “open” turns X into a topological space (X, T), and different choices of T can make the same underlying set behave very differently. On R^n, any norm induces a metric, and that metric induces a topology via ε-balls; although ε-balls change shape with the norm, the resulting topology on R^n stays the same. Not every topology comes from a metric: the indiscrete topology has only ∅ and X as open sets and cannot be metric-induced. This framework sets up later generalizations of closed sets and compact sets.

What exact conditions must a collection T of subsets of X satisfy to be a topology?

T must contain ∅ and X. It must be closed under finite intersections: if U and V are in T, then U ∩ V is also in T (and repeating this gives closure under any finite number of intersections). It must be closed under arbitrary unions: if {A_i} is any family of sets in T indexed by any set I (even uncountable), then ⋃_{i∈I} A_i is in T.

Why does choosing a topology matter even when the underlying set X stays the same?

Because openness is defined by T. Two different topologies on the same X can declare different subsets to be open, which changes what “neighborhoods,” continuity, and convergence mean. The same points can therefore behave very differently depending on which subsets are treated as open.

How does a norm on R^n lead to a topology?

A norm ||·|| on a vector space gives a metric by measuring distance with the norm of the difference vector: distance between x and y is ||x − y||. Given that metric, a set A is open if for every point a in A there exists ε > 0 such that the ε-ball around a lies entirely inside A.

What stays the same when switching norms on R^n, and what changes?

The ε-balls’ shapes can change because they depend on the chosen norm. However, the induced notion of openness on R^n does not change: all norms on R^n produce the same standard topology.

What is the indiscrete topology, and why is it important here?

The indiscrete topology on X declares only two open sets: ∅ and X. It is extremely rough—intuitively, every point is “close” to every other point because there are no smaller open neighborhoods. It also serves as an example of a topology that does not come from a metric.

Review Questions

Given a set X, how would you test whether a proposed collection T ⊆ P(X) is a topology? List the three closure/containment requirements.
Explain how a metric induces a topology using ε-balls, and describe what it means for a set A to be open.
Why can the same set R^n have the same topology from different norms even though the geometric shapes of balls differ?

Key Points

1
A topology on X is a collection T of subsets of X that includes ∅ and X, is closed under finite intersections, and is closed under arbitrary unions.
2
A topological space is the pair (X, T), where “open set” means “member of T.”
3
The same underlying set X can support many different topologies, leading to different notions of closeness and limit behavior.
4
On R^n, any norm induces a metric, and that metric induces a topology via ε-balls.
5
Changing the norm changes the shape of ε-balls but not the resulting topology on R^n.
6
Some topologies are not metric-induced; the indiscrete topology (only ∅ and X open) is a key example.

Highlights

A topology is built from a power set: pick which subsets are “open,” then enforce union and intersection rules.

On R^n, every norm produces the same standard topology even though ε-balls look different.

The indiscrete topology shows how extreme openness choices can make “closeness” nearly meaningless.

Not all topologies arise from metrics—metric-induced structure is optional, not guaranteed.

Topics

Topological Spaces
Topology Axioms
Open Sets
Metric-Induced Topology
Indiscrete Topology