Bases and Subspaces

In this lesson you’ll learn to define a topology by a basis instead of listing every open set, check the two conditions a basis must satisfy, and work with the subspace topology, where the ambient space is always part of the question.

Listing every open set of $\mathbb{R}$ is hopeless: there are uncountably many. But every one of them is a union of open intervals, so the intervals determine the whole topology. That is the idea.

A basis for a topology $\tau$ on $X$ is a collection $\mathcal{B}\subseteq\tau$ such that every open set is a union of members of $\mathcal{B}$.

Equivalently, and more usefully in practice: for every open $U$ and every $x\in U$, there is a basis element $B$ with $x\in B\subseteq U$.

The left panel builds a two-piece open set as a union of overlapping intervals, with the covering computed rather than drawn. Every open subset of $\mathbb{R}$ can be produced this way, which is what makes the intervals a basis.

Usually you have a candidate family and want a topology from it.

A collection $\mathcal{B}$ of subsets of $X$ is a basis for some topology if

1. $\mathcal{B}$ covers $X$: every point lies in some basis element.
2. For $B_1,B_2\in\mathcal{B}$ and $x\in B_1\cap B_2$, there is $B_3\in\mathcal{B}$ with $x\in B_3\subseteq B_1\cap B_2$.

Then the topology generated is all unions of basis elements. Condition 2 is weaker than requiring $B_1\cap B_2$ itself to be in $\mathcal{B}$, and that flexibility is what makes bases convenient: open intervals satisfy it because the intersection of two intervals is an interval, while open discs in the plane satisfy it even though the intersection of two discs is not a disc.

Space	Basis
$\mathbb{R}$, standard	open intervals $(a,b)$
$\mathbb{R}$, standard (countable)	intervals with rational endpoints
$\mathbb{R}^n$	open balls, or open boxes
Metric space	open balls $B(x,r)$
Discrete	singletons $\{x\}$
$\mathbb{R}$, lower limit	half-open intervals $[a,b)$
Product $X\times Y$	products $U\times V$ of open sets

The rational-endpoint basis is worth noticing: it is countable, which makes $\mathbb{R}$ second countable and is what lets many analysis arguments enumerate cases.

A related notion: a local basis at $x$ is a family of neighbourhoods of $x$ such that every neighbourhood of $x$ contains one of them. In a metric space the balls $B(x,1/n)$ form a countable local basis, and that is precisely why sequences work so well in metric spaces.

A convenient test: $\tau_1\subseteq\tau_2$ if and only if for every $x$ and every basis element $B_1\in\mathcal{B}_1$ containing $x$, there is $B_2\in\mathcal{B}_2$ with $x\in B_2\subseteq B_1$.

This is how the lower limit topology is shown to be strictly finer than the standard one: inside any $(a,b)$ containing $x$ you can fit $[x, b)$, but inside $[0,1)$ there is no open interval containing 0.

If $A\subseteq X$, the subspace topology on $A$ is

$$\tau_A = \{\,U\cap A \ :\ U\in\tau_X\,\}$$

So the open sets of $A$ are exactly the traces of the open sets of $X$. This is the only sensible choice, and it is why a subset of a space is automatically a space.

The right panel of the figure shows the consequence that surprises people: intersecting the ambient open interval with $[0,1]$ produces a half-open set, which is open in $[0,1]$ and not open in $\mathbb{R}$.

“Open” is never a property of a set by itself. It always means open in a particular space, and forgetting to say which is the most common source of confusion in this material.

Some properties pass to subspaces automatically, and some do not.

Property	Inherited by every subspace?
Hausdorff	yes
Metrizable	yes
Second countable	yes
Compact	no - only by closed subspaces
Connected	no
Complete (metric)	no - only by closed subspaces

A property inherited by all subspaces is called hereditary. Compactness and connectedness are not hereditary, which is why $[0,1]$ is compact and connected while its subspace $\{0\}\cup(\frac12,1)$ is neither.

Closed subspaces are the well-behaved case, and that pattern recurs: a closed subspace of a compact space is compact, and a closed subspace of a complete metric space is complete.

Example 1: Verify that open intervals form a basis for $\mathbb{R}$.

Solution. Condition 1: every real number lies in some interval, for instance $(x-1,x+1)$ ✓

Condition 2: the intersection of two open intervals is either empty or an open interval, so it is itself a basis element and serves as $B_3$ ✓

Answer: a basis. ∎

Example 2: Verify that open discs form a basis for $\mathbb{R}^2$.

Solution. Condition 1 is clear. For condition 2, let $x\in D_1\cap D_2$. Each disc contains a small disc about $x$, say of radii $r_1$ and $r_2$, so the disc about $x$ of radius $\min(r_1,r_2)$ lies inside both.

Answer: a basis. ∎

Note the intersection of two discs is not a disc, which is exactly why condition 2 is stated with a containment rather than membership.

Example 3: Show the intervals with rational endpoints form a basis for $\mathbb{R}$.

Solution. Given open $U$ and $x\in U$, there is $(a,b)\subseteq U$ containing $x$. By density of $\mathbb{Q}$, choose rationals $p\in(a,x)$ and $q\in(x,b)$. Then

$$x\in(p,q)\subseteq(a,b)\subseteq U$$

Answer: a basis, and a countable one. ∎

A countable basis for an uncountable space. This is what second countability means, and it is used constantly in analysis.

Example 4: Find the subspace topology on $\{0\}\cup(1,2)$ inside $\mathbb{R}$.

Solution. Open sets are intersections with ambient open sets. Taking $V=(-\frac12,\frac12)$ gives $\{0\}$, so the singleton $\{0\}$ is open in this subspace.

The subspace is therefore disconnected, splitting into the clopen pieces $\{0\}$ and $(1,2)$. ∎

A singleton can be open in a subspace without being open in $\mathbb{R}$, which is worth internalizing.

Example 5: Is $[0,1)$ open in $[0,2]$?

Solution. Yes: $[0,1) = (-1,1)\cap[0,2]$, an ambient open set intersected with the subspace.

Is it closed in $[0,2]$? Its complement in $[0,2]$ is $[1,2]$, which equals $[1,3]\cap[0,2]$ and is the intersection with a closed set, hence closed. So $[0,1)$ is open and not closed in $[0,2]$.

$$\text{open in } [0,2]$$

∎

Example 6: Show the subspace topology on $\mathbb{Z}\subseteq\mathbb{R}$ is discrete.

Solution. For any integer $n$,

$$\left(n-\tfrac12,\ n+\tfrac12\right)\cap\mathbb{Z} = \{n\}$$

so every singleton is open in the subspace, and therefore every subset is.

Answer: discrete. ∎

A discrete subspace of a non-discrete space. The subspace topology can look nothing like the ambient one.

Example 7: Show compactness is not hereditary.

Solution. $[0,1]$ is compact, as we prove in lesson 10. Its subspace $(0,1)$ is not: the cover by $\left(\frac1n,1\right)$ for $n\ge2$ has no finite subcover, since any finite collection has a smallest $\frac1n$ and misses everything below it.

Answer: compact space, non-compact subspace. ∎

Closed subspaces are the exception: a closed subspace of a compact space is always compact, and $(0,1)$ is not closed in $[0,1]$.

Constraint surfaces in robotics. A robot arm’s reachable set is a subspace of its full configuration space, and it inherits its topology. Path planning happens in that subspace, where openness and connectedness may differ from the ambient space.

Manifolds and data. The manifold hypothesis in machine learning says high-dimensional data lies near a low-dimensional subspace, and the subspace topology is what makes local charts and dimensionality reduction meaningful.

Level sets in simulation. A constraint surface such as an isosurface of temperature is studied with its subspace topology, and whether it is connected determines whether a physical region is isolated.

Sampling and second countability. A countable basis is what justifies approximating a continuous space by a countable grid, which every numerical method relies on. Without it, discretization has no guarantee of capturing the topology.

Digital signal spaces. The set of realizable filter coefficients is a subspace of a parameter space, and the design problem is constrained to it. Boundary behaviour differs between the two spaces, which is where marginal stability lives.

Geographic information systems. A region’s boundary is treated differently depending on whether you work in the plane or in the region as its own space, and inconsistent choices produce the classic sliver-polygon errors.