Theory

Quotients, gluing

Quotient topology

For any partition of $\mathcal{P}$ of $X$ into cells $Y_{\alpha }$, $\alpha \in \mathcal{A}$, there is a quotient topology structure on the set of cells $Y=\{Y_{\alpha }{\}}_{\mathcal{A}}$.

The open sets of $Y$ are the collections of cells whose unions are open in $X$.

A quotient space is therefore given by ‘collapsing’ some sets of points in $X$ into single objects $Y_{\alpha }$, and the open sets come from open sets in $X$ that are compatible with the partition, meaning they do not cross partition boundaries, or that they contain only whole cells.

There are many equivalent formulations of this notion, and these two are most common:

• Using an equivalence relation $\sim$ on $X$, whereby the equivalence classes $[x]$ determine cells of the partition. Then the quotient space is written $X/\sim$.
  • Recall that partitions and equivalence relations are interchangeable notions.
• Using a surjective function $f:X\twoheadrightarrow Y$, whereby the preimage fibers $f^{-1}(y)$ determine cells of the partition. Then $Y$ has the quotient topology when open sets are given by $V$ where $f^{-1}(V)$ is open in $X$.

Gluing

Two spaces $X_1$ and $X_2$ can be glued along specified subsets $A_1\subset X_1$ and $A_2\subset X_2$ using a bijection $\phi :A_1\to A_2$ by defining the quotient topology $X_1\bigsqcup X_2/\sim$ where $\sim$ identifies $a_1\in A_1$ with $\phi (a_1)\in A_2$.

Embedding, limit points, closure, boundary

Embedding

An embedding $f:A\hookrightarrow Y$ is a continuous injective map that induces a homeomorphism $f:A\to f(A)$.

It is important to emphasize that $f(A)$ should have the subspace topology inherited from $Y$.

For example, a knot in ${\mathbb{R}}^3$ is an embedding ${\mathbb{S}}^1\hookrightarrow {\mathbb{R}}^3$.

Embedding, limit points, closure, boundary (part 2)

Limit points

Given a subspace $S\subset X$, a point $p\in S$ is a limit point of $S$ when every open set around $p$ touches $S$ somewhere besides $p$.

$$p\in U\text{open}⟹(U\cap S)\backslash \{p\}\ne \varnothing$$

Notes:

• Limit points are sometimes called accumulation points.
• The limit point $p$ might be outside $S$!
• In a metric space, this can be reformulated: $p$ is a limit point of $S$ when there is some sequence $x_n\in S$ such that $d(x_n,p)\to 0$ as $n\to \infty$. This is written $x_n\to p$.
  • In other words, $p$ is “approached” from inside $S$.

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Interior and exterior

The interior of a set $S\subset X$ is the set of points $p\in S$ with an open neighborhood contained in $S$.

The exterior of a set $S\subset X$ is the interior of the complement of $S$.

Closure of a set

The closure $\overline{S}$ of a set $S\subset X$ is the union of $S$ with all limit points of $S$.

• Equivalently: the closure $\overline{S}$ is the complement of the union of all open sets in $X$ that don’t touch $S$.
• Equivalently: the closure $\overline{S}$ is the intersection of all closed sets containing $S$.

Boundary of a set

The boundary $\mathrm{Bd}(S)$ of a set $S\subset X$ is the set of points in both $\overline{S}$ and $\overline{(X\backslash S)}$.

• Equivalently: the boundary $\mathrm{Bd}(S)$ is the set of points for which every open neighborhood touches both the interior and the exterior of $S$.
• Boundary points are not always in $S$!