Notes

Introduction

This chapter generalises the concepts of connectedness, which is depended on by a basic theorem regarding continuous functions called the 'intermediate value theorem, as well as compactness, which sees application in establishing important results such as extreme value theorem and uniform continuity, from real numbers which is central in mathematical analysis to arbitrary topological spaces.

Connectedness

Connected Spaces

Definition. Let $X$ be a topological space. A separation of $X$ is pair $U, V$ of disjoint nonempty open subsets of $X$ whose union is $X$. The space is $X$ is said to be connected if there does not exist a separation of $X$.

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• If $X$ is connected, then so is any space homeomorphic to $X$, as $X$ is a topological property.

• Another way of phrasing the definition of connectedness is the following:

  A space $X$ is connected if and only if the only subsets of $X$ that are both open and closed in $X$ are the empty set and $X$ itself.

  This is because if $A$ is a nonempty proper subset of $X$ that is both open and closed in $X$, then the sets $U=A$ and $V=X \backslash A$ form a separation of $X$, since they are open, disjoint, nonempty, and have $X$ as their union. Conversely, if $U$ and $V$ form a separation of $X$, then $U$ is nonempty and different from $X$, and it is both open and closed in $X$.

[Lemma 23.1] (Subspace connectedness). If $Y$ is a subspace of $X$, a separation of $Y$ is a pair of disjoint nonempty sets $A$ and $B$ whose union is $Y$, neither of which contains a limit point of the other. The space $Y$ is connected if there exists no separation of $Y$.

Constructing new connected spaces from old.

[Lemma 23.2] If the sets $C$ and $D$ form a separation of $X$, and if $Y$ is a connected subspace of $X$, then $Y$ lies entirely within either $C$ or $D$.

[Theorem 23.3] The union of a collection of connected subspaces of $X$ that have a point in common (i.e. the intersection of the collection is nonempty) is connected.

[Theorem 23.4] Let $A$ be a connected subspace of $X$. If $A \subseteq B \subseteq \overline{A}$, then $B$ is also connected.

Equivalently: if $B$ is formed by adjoining to the connected subspace $A$ some or all of tis limit points, then $B$ is connected.

[Theorem 23.5] (Continuous function preserve connectedness). The image of a connected space under a continuous map is connected.

[Theorem 23.6] A finite Cartesian product of connected spaces is connected.

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Does Theorem 23.6 extend to arbitrary products of connected space? The answer is: it depends on the topology used for the product.

In fact, an arbitrary product of connected space is connected in the product topology. A proof of this result is a solution to Exercise 23.10.

Connected Subspaces of the Real Line

Definition. A simply ordered (a.k.a. strictly totally ordered) set $L$ having more than one element is called a linear continuum if the following hold:

1. $L$ has the least upper bound property.
2. If $x < y$, there exists $z$ such that $x < z < y$ (i.e. the order on $L$ is dense).

[Theorem 24.1] If $L$ is a linear continuum in the order topology, then $L$ is connected, and so are intervals and rays in $L$.

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Since $\mathbb{R}$ is a linear continuum in order topology, we immediately see that:

[Corollary 24.2] The real line $\mathbb{R}$ is connected and so are intervals and rays in $\mathbb{R}$.

One may also ask: does the converse of Theorem 24.1 hold? The answer is in fact yes, and a proof of this is a solution to Exercise 24.4.

[Theorem 24.3] (Intermediate Value Theorem). Let $f : X \to Y$ be a continuous map, where $X$ is a connected space and $Y$ is an ordered set in the order topology. If $a$ and $b$ are two points of $X$ and if $r$ is a point of $Y$ lying between $f(a)$ and $f(b)$, then there exists a point $c$ of $X$ such that $f(c)=r$.

Definition. (Path-connectedness). Given points $x$ and $y$ of the space $X$, a path in $X$ from $x$ to $y$ is a continuous map $f : [a,b] \to X$ of some closed interval in the real line into $X$, such that $f(a)=x$ and $f(b)=y$. A space $X$ is said to be path connected if every pair of points of $X$ can be joined by a path in $X$.

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The converse of the definition above does not hold; a connected space need not be path connected.

[Example 6]. The orderred square $I_o^2$ (recall that it is defined as $I \times I \subset \mathbb{R}^2$, where $I := [0,1]$, endowed with the dictionary order) is connected but not path connected.

Since it is a linear continuum (as a subset of $\mathbb{R}^2$), it is connected. We show that $I_o^2$ is not path connected as follows:

Let $p = (0,0)$ and $q = (1,1)$ (note that they are respectively the smallest and largest element of $I_o^2$). Assume for the sake of contradiction that there is a path $f : [a,b] \to I_o^2$ joining $p$ and $q$. It follows that the image set $f([a,b])$ must contain every point $(x,y)$ of $I_o^2$ by the intermediate value theorem (recall that a path is a continuous map).

Therefore, for each $x \in I$, the set

$$U_x = f^{-1}(\{x\} \times (0,1))$$

is a nonempty subset of $[a,b]$; by continuity, it is open in $[a,b]$. Choose, for each $x \in I$, a rational number $q_x$ belonging to $U_x$. Since the sets $U_x$ are disjoint, the map $x \mapsto q_x$ is an injective mapping of $I$ into $\mathbb{Q}$. This contradicts the fact that the inverval $I$ is uncountable.

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The uncountability of $I$ comes from the fact that if there were a bijective mapping from the open interval $(0,1)$ (note that this is just $I$ but 0 and 1 are excluded) to the natural numbers (which is the definition of being countable) so that each element of it can be indexed as $x_i=0.d_{i1}d_{i2}d_{i3}\cdots$ where $i \in \mathbb{N}$, then one can come up with an element in $I$ given by $x=0.a_1 a_2 a_3 \cdots$ where $a_i=d_{ii}+1 \pmod{10}$, which is not equal to any of the $x_i$'s, contradicting the existence of said bijective mapping.

Components and Local Connectedness

Breaking up into smaller pieces.

The notion of local connectedness refers to a natural process of breaking up an arbitrary (topological) space into connected components.

Definition. ([Connected] components). Given $X$, define an equivalence relation on $X$ by setting $x \sim y$ if there is a connected subspace of $X$ containing both $x$ and $y$. The equivalence classes are then called the components (or the 'connected components') of $X$.

[Theorem 25.1] The components of $X$ are connected disjoint subspaces of $X$ whose union is $X$, such that each nonempty connected subspace of $X$ intersects only one of them.

This comes from the fact that the components form equivalence classes in $X$, which induces a partition of $X$.

Definition. (Path components). We define another equivalence relation on the space $X$ by defining $x \sim y$ if there is a path in $X$ from $x$ to $y$. The equivalence classes are called the path components of $X$.

[Theorem 25.2] The path components of $X$ are path-connected disjoint subspaces of $X$ whose union is $X$, such that each nonempty path-connected subspace of $X$ intersects only one of them.

Definition. (Local connectedness). A space $X$ is said to be locally connected at $x$ if for every neighbourhood $U$ of $x$, there is a connected neighbourhood $V$ of $x$ contained in $U$. If $X$ is a locally connected at each of its points, it is said simply to be locally connected.

Similarly, a space $X$ is said to be locally path connected at $x$ if for every neighbourhood $U$ of $x$, there is a path-connected neighbourhood $V$ of $x$ contained in $U$. If $X$ is locally path connected at each of its points, then it is said to be locally path connected.

[Theorem 25.3] A space $X$ is locally connected if and only if for every open set $U$ of $X$, each component of $U$ is open in $X$.

[Theorem 25.4] A space $X$ is locally path connected if and only if for every open set $U$ of $X$, each path component of $U$ is open in $X$.

[Theorem 25.5] (Relation between path components and components). If $X$ is a topological space, each path component of $X$ lies in a component of $X$. If $X$ is locally path connected, then the components and the path components of $X$ are the same.

Compactness

Definition. A collection $\mathcal{A}$ of subsets of a space $X$ is said to cover $X$, or to be a covering of $X$, if the union of the elements of $A$ is equal to $X$. It is called an open covering of $X$ if its elements are open subsets of $X$.

Definition. A space $X$ is said to be compact if every open covering $\mathcal{A}$ of $X$ contains a finite subcover (i.e. a finite subcollection that also covers $X$).