Open sets and neighborhoods
If open sets are responsible for nearness, continuity, and topological invariants, what rules must they satisfy?
How can local room replace distance?
Once distance stops being primary, a space still needs a way to say which points are locally close, which regions can be observed from inside, and which changes preserve the arrangement of points.
Open sets answer that need. An open set is a region that has room around each of its points. If a point lies in an open region, then the point is not sitting on an included edge of that region; it can move a little while still remaining inside.
On the real line, the interval (0, 1) is open. Every point inside it has some small interval around it that remains inside (0, 1). The point 1/2 has plenty of room. The point 0.99 has less room, but still has some. The endpoints 0 and 1 are not included, so the region never has to provide room around them.
0 1/2 1
o---------*---------o
inside points have local room
endpoints are not included
That example uses the ordinary distance on the line to motivate the idea. Topology keeps the idea and abstracts away the measuring device.
A topology on a set X is a chosen collection of subsets of X, called open sets, satisfying three rules:
- The empty set and the whole set
Xare open. - Any union of open sets is open.
- Any finite intersection of open sets is open.
A topological space is a set together with such a chosen collection of open sets.
set of points X
+
chosen open subsets
=
topological space
These rules are minimal local bookkeeping. They say how open regions behave when we combine observations. The whole space is open because every point has all the room the space can provide. The empty set is open because it contains no point that could fail the room condition. These two cases give the structure harmless endpoints.
Unions express alternatives. If a point lies in a union of open regions, then it lies in at least one of them. That one region already supplies local room around the point, so the union supplies local room too.
open U open V
\ /
\ /
U union V
choose the open region containing the point
We may combine any number of open alternatives and still have an open region. Local room is inherited from one successful alternative.
Finite intersections express simultaneous requirements. If a point lies in U and also in V, and both are open, then each gives some room around the point. We can choose room small enough to fit inside both.
require U
require V
|
v
require U intersection V
This works for a fixed finite number of requirements. With two, three, or any finite number of open conditions, we can shrink the local room until all of them are satisfied at once.
Infinite intersections are different. On the real line, each interval (-1/n, 1/n) is open. Their intersection over all positive integers is the single point {0}. The single point has no interval of room around it, so it is not open in the usual topology.
(-1, 1)
(-1/2, 1/2)
(-1/3, 1/3)
...
intersection = {0}
Finitely many local constraints can be satisfied by shrinking room. Infinitely many constraints may shrink all room away.
The three axioms therefore preserve the basic meaning of openness:
open regions are stable under arbitrary alternatives and finite simultaneous tests.
This is the structure that replaces distance.
Neighborhoods give the pointwise version of the same idea. A neighborhood of a point x is a region that contains some open set around x. It may itself be open, or it may be larger than an open region. What matters is that it gives x local room.
x in O and O inside N
O = open room around x
N = neighborhood of x
Open sets and neighborhoods determine each other. If we know the open sets, we know the neighborhoods: a set is a neighborhood of x when it contains an open set containing x. If we know the neighborhoods of every point and they behave coherently, we can recover the open sets: a region is open when it is a neighborhood of each of its points.
So there are two complementary views:
open-set view:
regions first
neighborhood view:
local room around points first
The open-set view is efficient for defining continuity. The neighborhood view is closer to intuition: it says what each point can locally see. Topology is placed on a set. The set alone does not determine its open sets.
The same underlying points can carry many different topologies. One extreme is the discrete topology, where every subset is open. In a discrete space, each point has complete local separation from every other point because {x} is open for every point x.
discrete topology:
every subset is open
each point can be isolated
Another extreme is the indiscrete topology, where only the empty set and the whole set are open. In an indiscrete space, the topology has almost no local resolving power. It cannot distinguish one point from another by smaller open regions.
indiscrete topology:
only empty set and X are open
points cannot be locally separated
Between these extremes lie the familiar spaces of analysis and geometry. A metric space has a distance function, and that distance creates open sets: a set is open when every point inside it contains some open ball still inside the set.
metric distance
|
v
open balls
|
v
open sets
Topology keeps the open sets and forgets the exact distances that generated them. This is a genuine loss of information. If the usual distance on the real line is doubled, the actual numbers assigned to distances change. The open sets remain the same. Topology treats those two metric descriptions as having the same local structure.
It is also a genuine gain. Once the open sets are isolated, continuity can be studied without choosing a particular measurement.
The topology determines which maps count as continuous.
A map f: X -> Y is continuous when every open set in Y pulls back to an open set in X.
X ----f----> Y
| |
| open | open
| preimage | region
v v
Open sets are observable regions or local tests. Continuity says that every target test can be translated into a source test of the same kind. The source has enough local structure to control the target condition.
In neighborhood language, the same idea says: whenever f(x) is required to lie in a target neighborhood, there is a source neighborhood of x whose points all satisfy that requirement after applying f.
source neighborhood of x
|
| f
v
target neighborhood of f(x)
The open-set definition gives the concise global test. The neighborhood definition gives the local meaning. They are two descriptions of the same structural requirement.
If the source has the discrete topology, every map out of it is continuous, because every inverse image is automatically open. If the target has the indiscrete topology, every map into it is continuous, because there are only two open sets to check. These examples are extreme, but they show that continuity belongs to the chosen topologies rather than to the underlying point functions alone.
function alone:
point assignment
function + topologies:
continuity question
A point function can be continuous for one choice of topology and fail to be continuous for another. The topology decides which local distinctions matter.
This is the main lesson. Open sets carry the local information of the space. Once that local information is fixed, continuity becomes a structural condition rather than a numerical estimate.
They also begin to shape the invariants of topology. If a feature can be described using only the pattern of open regions, then it has a chance to survive continuous change. The present move is simpler: open sets provide the local language in which those later questions can be asked.
In metric analysis, nearness is measured directly. In topology, nearness is encoded by the pattern of open neighborhoods. A point is understood by the open regions around it. A map is understood by how it translates open regions backward. A property is topological when it can be expressed using this open-set structure and survives continuous change.
metric:
distance first
topology:
open neighborhoods first
This gives topology its flexibility. The same metric idea can be recovered when a metric is available, but topology can also describe spaces where no particular metric is chosen, where many metrics give the same open sets, or where local structure is the only thing that matters.
There is still a boundary. Open sets tell us what continuity means, but they do not yet tell us when two spaces should count as the same. We need a notion of equivalence that preserves the open-set structure in both directions.
The objects are topological spaces: sets equipped with chosen open subsets. The allowed transformations are continuous maps, which pull open sets back to open sets. What composes are continuous maps, and composition is coherent because inverse images compose. The invariant pattern is local organization: which regions count as open, which neighborhoods surround each point, and which maps preserve that structure. The defining relation is the topology itself: a collection of open sets closed under arbitrary unions and finite intersections, containing the empty set and the whole space. Equivalence is still missing.
Open sets are the local grammar of space after distance has been removed.
Continuity forced open sets into view. Open sets now reveal what topology stores: not lengths, angles, or coordinates, but the pattern of local regions and the way those regions fit together.
When do two spaces have the same topology, even if their points are named, drawn, or measured differently?