7 Topology[0G5]

Let $X$ be a fixed and non-empty set. We will use this notation. For each set $A \subseteq X$ we define that $A^{c} = X ⧵ A$ is the complement to A.

Definition 241

[2DY] A topological space is a pair $(X, 𝜏)$ where $X$ is a non-empty set with associated the family $𝜏$ of the open sets, which is called topology.

Definition 242

[0G6] A topology $𝜏 \subseteq P (X)$ is a family of subsets of $X$ that are called open sets. This family enjoys three properties: $\emptyset, X$ are open; the intersection of a finite number of open sets is an open sets; the union of an arbitrary number of open sets is an open set.

A set $A$ is closed if $A^{c}$ is open.

Definition 243

[0G7] Let $A, B \subseteq X$ be two subsets.

The interior of $A$ , denoted by $A^{\circ}$ , is the union of all the open sets contained in $A$ , and therefore is the biggest open set contained in $A$ ;
the closure of $B$ , denoted by $\overset{―}{B}$ , is the intersection of all the closed sets that contain $B$ , i.e. is the smallest closed that contains $B$ .
We say that $A$ is dense in $B$ if $\overset{―}{A} \supseteq B$ . ¹
The boundary $\partial A$ of $A$ is $\partial A = \overset{―}{A} ⧵ A^{\circ}$ .

Definition 244

[0G8] A topological space $(X, 𝜏)$ is said to be $T_{2}$ , or ”Hausdorff space”, if $\forall x, y \in X$ exist $U, V \in 𝜏$ open disjoint with $x \in U, y \in V$ .

Definition 245

[2F6]Any set $X$ can be endowed with many different topologies. Here are two simple examples:

When a set $X$ is endowed with the discrete topology, then all sets are open, and therefore closed. Equivalently, the discrete topology is caracterized by: every singleton is an open set.
When a set $X$ is endowed with the indiscrete topology, then the only open (and, closed) sets are $X, \emptyset$ .

Further informations on these subjects may be found in Chap. 2 of [ 22 ] or in [ 14 ] .

Remark 246

[2DH]A metric space is a special case of topological space, because the open subsets of the metric space satisfy the Definition 242; the associated topology is always Hausdorff. The following results therefore also apply to metric spaces.

E246

[0G9]Show that if the space is $T_{2}$ then every singleton ${x}$ is closed.

E246

[0GB]Show that if $A \subseteq B$ then $\overset{―}{A} \subseteq \overset{―}{B}$ and $A^{\circ} \subseteq B^{\circ}$

E246

[0GC]Show that if $A = B^{c}$ then $(\overset{―}{B})^{c} = A^{\circ}$ , using the definitions 242 and 243.

E246

[0GD]Note that $A \supseteq A^{\circ}$ and $B \subseteq \overset{―}{B}$ , generally. Show that $A$ is open if and only if $A = A^{\circ}$ ; and that $B$ is closed if and only if $B = \overset{―}{B}$ , using definitions 242 and 243.

E246

[0GF] Topics:interior. Given $X$ , a topological space, and $A \subseteq X$ , show that

A^{\circ} = {(A^{\circ})}^{\circ} .

using the definition of $A^{\circ}$ given above.

(For the case of $X$ metric space, see also 10)

Hidden solution: [UNACCESSIBLE UUID ’0GG’]

E246

[0GH] Topics:closing. Given $X$ topological space and $A \subseteq X$ , show that

\overset{―}{A} = \overset{―}{(\overset{―}{A})}

or by switching to complement with respect to 5, and using the definition of $\overset{―}{A}$ like ”intersection of all the locks they contain $A$ ”.

(For the case of $X$ metric space, see also 13)

E246

[0GJ] Topics:closure, interior. Let $X$ be a topological space and $A \subseteq X$ open.

Show that $A \subseteq {(\overset{―}{A})}^{\circ}$ (the interior of the closure of $A$ ).
Find an example of an open set $A \subset ℝ$ for which $A \neq {(\overset{―}{A})}^{\circ}$ .
Then formulate a similar statement for $A$ closed, transitioning on to the complement.

Hidden solution: [UNACCESSIBLE UUID ’0GK’]

E246

[0GM]Given the sets $A, B \subseteq R$ , determine the relations between the following pairs of sets

\begin{aligned} \overset{―}{A \cup B} \text{and}~ & \overset{―}{A} \cup \overset{―}{B}, \\ \overset{―}{A \cap B} \text{and}~ & \overset{―}{A} \cap \overset{―}{B}, \\ {(A \cup B)}^{\circ} \text{and}~ & A^{\circ} \cup B^{\circ}, \\ {(A \cap B)}^{\circ} \text{and}~ & A^{\circ} \cap B^{\circ} . \end{aligned}

[UNACCESSIBLE UUID ’0GN’]

Hidden solution: [UNACCESSIBLE UUID ’0GP’] [0GQ] Prerequisites:55,53,3.Difficulty:*.(Replaces 29W)

Let $(X, τ)$ be a topological space. Consider the descending ordering between sets ² , with this ordering $𝜏$ is a directed set; we note that it has minimum, given by $\emptyset$ .

Now suppose the topology is Hausdorff. Then taken $x \in A$ , let $U = {A \in 𝜏 : x \in A}$ be the family of the open sets that contain $x$ : show that $U$ is a directed set; show that it has minimum if and only if the singleton ${x}$ is open (and in this case the minimum is ${x}$ ). ³

Hidden solution: [UNACCESSIBLE UUID ’0GR’]

By the exercise 3, when ${x}$ is not open then $U$ is a filtering set, and therefore can be used as a family of indices to define a nontrivial ”limit” (see Remark 239). We will see applications in section 7.7.

[0GS]Note:Written exam of 25 March 2017.Let

(X, τ)

(Y, θ)

be two topological spaces with non-empty intersection and assume that the topologies restricted to

C = X \cap Y

coincide (i.e.

τ_{| C} = θ_{| C}

) ⁴ and that

C

is open in both topologies (i.e.

C \in τ, C \in θ

). Prove that there is only one topology

σ

Z = X \cup Y

such that

σ_{| X} = τ

and

σ_{| Y} = θ

and that

X, Y \in σ

. Hidden solution: [UNACCESSIBLE UUID ’0GT’][UNACCESSIBLE UUID ’0GV’]