2 Topologies, Bases, Subspaces and Products

28 January 2026 · Denis Auroux

Remark (Recall).

A metric space is a set with a distance function such that (1) iff ; (2) ; (3) .
Open balls: . is open iff such that .
is continuous such that open, is open. This will be the definition outside of the metric case.
Limits of sequences: in if such that . every open subset contains for all but finitely many . This will be the definition of limit outside the metric case.

Interlude: what is ?

Answer: it’s an ordered field (i.e. and compatible with the usual rules) with the least upper bound property: every nonempty subset that admits an upper bound ( such that , ) has a least upper bound (i.e. is an upper bound, and every upper bound for is ).

The l.u.b. property is equivalent to completeness of ; any ordered field with this property is isomorphic to . Constructions of from involve adding the missing elements (irrationals) so that the l.u.b. property / completeness holds; the elements of end up being either the sups of certain subsets of (see e.g. Rudin) or the limits of Cauchy sequences in (see HW).

Topological spaces

We will now reformulate/generalize the above notions in the context of topological spaces, i.e. sets equipped with a topology which may or may not come from a metric.

Definition 2.1. A topological space is a set together with a collection , the open sets in , such that:

, ;
arbitrary unions of open sets are open;
finite intersections of open sets are open.

Why bother? One answer: many natural topologies do not come from a metric! E.g., in analysis:

On the space of (bounded) functions , the uniform convergence topology ( iff ) comes from a metric (), but pointwise convergence ( iff , ) doesn’t (“product topology”).
The topology on smooth functions doesn’t come from a metric either.

And on the other hand, a metric contains extraneous information for topology. E.g. , , have the same open sets, hence the same topology.

Definition 2.2.

is continuous if , open is open.
A sequence in converges to a limit () if open, such that .

Example 2.3.

If is a metric space, then is the metric topology.
The discrete topology: (every subset is open and closed). This is in fact a metric topology: set . (E.g. the usual topology on .)

These abstract definitions imply basic facts about continuity, such as:

Proposition 2.4.

If is continuous and in , then in .
and continuous continuous.

Exercise 2.5. Check this.

Finer and coarser topologies

Given two topologies on , if we say is finer and is coarser. The finest topology on is the discrete one (all points are isolated), while the coarsest is (“one big clump”).

The finer topology has more open sets; it’s easier for functions to be continuous with respect to than (every function from a discrete set is continuous).
It’s harder for sequences to converge in . E.g. on a discrete set, convergent sequences must be constant after finitely many terms; while for every sequence converges to every point of ; in particular the limit isn’t unique!

Bases

Keeping track of all the open sets is cumbersome. In metric spaces we started with open balls and got a characterization of open sets in terms of these. The analogous notion for a general topology is that of basis.

Definition 2.6. Assume is a collection of subsets of such that

;
if and , then such that .

Then we say is a basis and generates the topology = arbitrary unions of elements of . Equivalently: such that .

Exercise 2.7. Check: (1) the two characterizations of are equivalent; (2) is a topology.

Remark. Unlike bases in linear algebra, bases in topology can contain redundant info; a better analogy is with generating sets. E.g. the metric topology is generated by any of:

all open sets;
open balls , , ;
open balls , ;
or just open balls , , where is a dense subset (every nonempty open set intersects ), e.g. .

So for example the usual topology on or actually admits a countable basis!

Making new topological spaces: subspaces, products

Definition 2.8. Let be a topological space and any subset. The subspace topology on is (Verify: this satisfies the axioms of a topology.)

Warning: it’s important when stating “ is open” to be clear: as a subset of what space? E.g. is always open as a subset of itself! is open in but not in .

The subspace topology is the coarsest topology on that makes the inclusion continuous (HW). Also, if comes from a metric on , then comes from .

Notes not reached in this lecture, covered at the start of Lecture 3

Definition 2.9. Let , be topological spaces. The product topology on is the topology generated by the basis

(Check that is a basis. Why isn’t already a topology?)

When are metric spaces, this is also a metric topology, defined e.g. by (check! Key observation: ). Or in fact define the same topology on (see HW for the case of ). So this gives the usual topology on .
In general, it’s the coarsest topology on such that the projection maps and are continuous (HW!). Also: iff and .
Similarly for finite products . For infinite products there are several different natural topologies; see next week.

Homeomorphisms

What is the correct notion of two topological spaces being “the same”?

Definition 2.10. are homeomorphic if there exist continuous maps and such that , .

Equivalently, a homeomorphism is a continuous bijection such that is continuous. I.e., a bijection under which .

Remark.

A continuous bijection need not be a homeomorphism. E.g. take with two topologies, strictly finer than . Then is a bijection, continuous since , but not a homeomorphism.
Say a metric space is bounded if . This is not a topological property: e.g. , is a homeomorphism ( and are continuous), so is homeomorphic to (or any open interval in ).

Closed sets

Definition 2.11. A subset of a topological space is closed if is open.

Warning: subsets can be both closed and open, e.g. and , or neither (e.g. or in ).

The axioms of open sets imply:

, are closed;
arbitrary intersections of closed sets are closed;
finite unions of closed sets are closed.

Definition 2.12. Let be any subset. We define:

the closure of : = the smallest closed set containing (, and is closed since it’s an intersection of closed sets);
the interior of : = the largest open set contained in (open);
the boundary of : .

Example 2.13. , usual topology , , .

Remark.

is closed iff ; open iff .
and .

Definition 2.14. Say is a neighborhood of if is open and .

Proposition 2.15.

iff contains a neighborhood of .
iff every neighborhood of intersects nontrivially.

Proof. Check this! (1) follows from the definitions: open such that .

(2) follows from (1) and : open, . ∎

Definition 2.16. Say is dense if (i.e. every nonempty open subset of intersects nontrivially).

Example 2.17. is dense in (for the usual topology).