# Contents

## Definition

The point space $\ast$ is the topological space whose underlying set is the singleton, and equipped with the unique topology that this set carries.

## Properties

### General

The point space is the terminal object in the category Top of topological spaces.

For $X$ any topological space, then for every element of its underlying set there is a continuous function from the point space

$\ast \longrightarrow X$

whose image is that point, and every such continuous function arises this way

### Relation to irreducible closed subspaces

For the following we write the point space explicitly as

$\ast = \left\{ \{1\}, \, \tau_\ast = \left\{ \emptyset, \{1\} \right\} \right\}$
###### Proposition

For $(X,\tau)$ a topological space, then there is a bijection between the irreducible closed subspaces of $(X,\tau)$ and the frame homomorphisms from $\tau_X$ to $\tau_\ast$ from the frame of opens of $X$ to that of the point space. Moreover, this is given by

$\array{ Hom_{Frame}(\tau_X, \tau_\ast) &\underoverset{\simeq}{}{\longrightarrow}& IrrClSub(X) \\ \phi &\mapsto& X \backslash U_\emptyset(\phi) }$

where $U_\emptyset(\phi)$ is the union of all elements $U \in \tau_x$ such that $\phi(U) = \emptyset$:

$U_{\emptyset}(\phi) \coloneqq \underset{{U \in \tau_X} \atop {\phi(U) = \emptyset} }{\cup} U \,.$

###### Proof

First we need to show that the function is well defined in that given a frame homomorphism $\phi \colon \tau_X \to \tau_\ast$ then $X \backslash U_\emptyset(\phi)$ is indeed an irreducible closed subspace.

To that end observe that:

$(\ast)$ If there are two elements $U_1, U_2 \in \tau_X$ with $U_1 \cap U_2 \subset U_{\emptyset}(\phi)$ then $U_1 \subset U_{\emptyset}(\phi)$ or $U_2 \subset U_{\emptyset}(\phi)$.

This is because

\begin{aligned} \phi(U_1) \cap \phi(U_2) & = \phi(U_1 \cap U_2) \\ & \subset \phi(U_{\emptyset}(\phi)) \\ & = \emptyset \end{aligned} \,,

where the first equality holds because $\phi$ preserves finite intersections by def. , the inclusion holds because $\phi$ respects inclusions by remark , and the second equality holds because $\phi$ preserves arbitrary unions by def. . But in $\tau_\ast = \{\emptyset, \{1\}\}$ the intersection of two open subsets is empty precisely if at least one of them is empty, hence $\phi(U_1) = \emptyset$ or $\phi(U_2) = \emptyset$. But this means that $U_1 \subset U_{\emptyset}(\phi)$ or $U_2 \subset U_{\emptyset}(\phi)$, as claimed.

Now according to prop. the condition $(\ast)$ identifies the complement $X \backslash U_{\emptyset}(\phi)$ as an irreducible closed subspace of $(X,\tau)$.

Conversely, given an irreducible closed subset $X \backslash U_0$, define $\phi$ by

$\phi \;\colon\; U \mapsto \left\{ \array{ \emptyset & \vert \, \text{if} \, U \subset U_0 \\ \{1\} & \vert \, \text{otherwise} } \right. \,.$

This does preserve

1. arbitrary unions

because $\phi(\underset{i}{\cup} U_i) = \{\emptyset\}$ precisely if $\underset{i}{\cup}U_i \subset U_0$ which is the case precisely if all $U_i \subset U_0$, which means that all $\phi(U_i) = \emptyset$ and because $\underset{i}{\cup}\emptyset = \emptyset$;

while $\phi(\underset{i}{\cup}U_1) = \{1\}$ as soon as one of the $U_i$ is not contained in $U_0$, which means that one of the $\phi(U_i) = \{1\}$ which means that $\underset{i}{\cup} \phi(U_i) = \{1\}$;

2. finite intersections

because if $U_1 \cap U_2 \subset U_0$, then by $(\ast)$ $U_1 \in U_0$ or $U_2 \in U_0$, whence $\phi(U_1) = \emptyset$ or $\phi(U_2) = \emptyset$, whence with $\phi(U_1 \cap U_2) = \emptyset$ also $\phi(U_1) \cap \phi(U_2) = \emptyset$;

while if $U_1 \cap U_2$ is not contained in $U_0$ then neither $U_1$ nor $U_2$ is contained in $U_0$ and hence with $\phi(U_1 \cap U_2) = \{1\}$ also $\phi(U_1) \cap \phi(U_2) = \{1\} \cap \{1\} = \{1\}$.

Hence this is indeed a frame homomorphism $\tau_X \to \tau_\ast$.

Finally, it is clear that these two operations are inverse to each other.

examples of universal constructions of topological spaces:

$\phantom{AAAA}$limits$\phantom{AAAA}$colimits
$\,$ point space$\,$$\,$ empty space $\,$
$\,$ product topological space $\,$$\,$ disjoint union topological space $\,$
$\,$ topological subspace $\,$$\,$ quotient topological space $\,$
$\,$ fiber space $\,$$\,$ space attachment $\,$
$\,$ mapping cocylinder, mapping cocone $\,$$\,$ mapping cylinder, mapping cone, mapping telescope $\,$
$\,$ cell complex, CW-complex $\,$

Last revised on May 9, 2017 at 03:52:52. See the history of this page for a list of all contributions to it.