Classnotes for September 22

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Exams Apr 24 Final, PM

Classnotes for September 22

The Induced Subset Topology

Given a space X with a topology $\mathcal{T}$, there is a natural way to induce a topology $\mathcal{T}^\prime$ on a subset $Y \subset X$ that is in many ways compatible with $\mathcal{T}$. We would expect of such a natural topology certain nice properties such as:

1. The map which embeds Y into X is continuous, i.e. $\iota\colon Y \rightarrow X$ is continuous.
2. A function f from another space Z to Y continuous whenever $g\colon Z \rightarrow X$ defined as $g = \iota \circ f$.

This first condition puts a lower bound, so to speak, on the fineness of $\mathcal{T}^\prime$: it must have at least the open sets that $\mathcal{T}$ grants Y. The second condition gives an upper limit on the fineness in a similar way. The following theorem demonstrates that these 'bounds' agree and yield the sought-after natural subset topology.

Theorem: If X is a topological space and $Y \subset X$, there exists a unique topology $\mathcal{T}_Y$ on Y such that:

1. $\iota\colon Y \rightarrow X$ is continuous
2. $f\colon Z \rightarrow Y$ is continuous whenever $g\colon Z \rightarrow X$ defined as $g = \iota \circ f$ is continuous

and it is explicitly described as: $\mathcal{T}_Y = \{U \cap Y: \forall\;open\; U \subset X\}$

The proof of this theorem is left as an easy exercise. The induced subset topology can be thought of as restricting a topology to a subset.

If X and Y are topological spaces and $U \subset X$ and $V \subset Y$, then the set $U \times V \subset X \times Y$ could be endowed with two potentially distinct natural topologies. One being as a product of subsets (i.e. endow U and V with subset topologies, as described above, from X and Y respectively and then give $U \times Y$ the product topology) or as a subset of a product (i.e. endow $X \times Y$ with the product topology and give $U \times V \subset X \times Y$ the induced subset topology). However, as one might expect, these two processes yield identical topologies.

There are some further properties of induced product and subset topologies worth noting:

Theorem:

1. Taking product topologies is associative, i.e. $(X \times Y) \times Z = X \times (Y \times Z)$
2. If $Z \subset Y \subset X$ with X being a topological space, then the subset topology induced on Z by considering it as a subset of Y (with a subset topology induced from X) is equal to the subset topology induced on Z by considering it directly as a subset of X.
3. $X \times Y$ is homeomorphic (though not generally equal) to $Y \times X$.
4. If $(a,b) \subset X$ is an interval within an ordered set X, then the interval topology induced on (a,b) by its ordering is the same as the subset topology induced from X with its interval topology.

Closed Sets

Defintion: A subset F of a topological space X is called closed if $F^c = {X \setminus F} = \{x \in X : x \notin F\}$ is open.

Closed sets have properties that are strikingly similar to those of open sets:

Theorem: If X is a topological space then:

1. $X, \empty$ are both closed.
2. An arbitrary intersection of closed sets is a closed set.
3. The finite union of closed sets is a closed set.

Proof:

1. $X^c = \empty$ and $\empty^c \ X$ are both open.
2. $(\cap_{(\alpha \in I)} F_\alpha)^c = (\cup_{(\alpha \in I)} (F_\alpha)^c)$ by de Morgan's Law. This set is the union of open sets and is therefore open. So, its complement $(\cap_{(\alpha \in I)} F_\alpha)$ is closed.
3. $(\cup_{(i\in I)} F_i)^c = (\cap_{(i \in I)} (F_i)^c)$ by de Morgan's Law. This set is the finite intersection of open sets and is therefore open. So its complement $(\cup_{(i\in I)} F_i)$ is closed.

Note that a set can be both closed and open; the empty set and the entire space, for example, are always both closed and open. Also, note that the definition of a topology could just as well have used closed sets rather than open sets. The choice of open sets as the basic elements of topology was arbitrary.

Interior and Closure

We will introduce two very useful operators on sets that are topological in nature:

Definition: Let $A \subset X$ where X is a topological space.

1. The closure of X, denoted by Cl(A), is defined as the smallest closed set containing A, or equivalently, the intersection of all closed subsets of X containing A.
2. The interior of X, denoted by Int(A), is defined as the largest open set contained in A, or equivalently, the union of all open subsets of X contained in A.

As an exercise, one many show that arbitrary applications of the interior, closure, and complement operators to a subset A of a topological space can yield, at most, 14 distinct sets. Also, one can find a subset of $\mathbb{R}$ which yields 14 distinct sets by applying these operators in different sequences. Curious or lazy students can find the proof and example here.

The interior and closure of a set have very useful characterisations:

Theorem:

1. $x \in Int(A)$ if and only if there exists an open set U such that $x \in U \subset A$.
2. $x \in Cl(A)$ if and only if for every open set U such that $x \in U$, $\;U \cap A \ne \empty$.

Proof:

1. If $x \in U \subset A$, then x is in the union of all open subsets of A, and is therefore in Int(A). Conversely, if $x \in Int(A)$ then Int(A) is open and $x \in Int(A) = U \subset A$.
2. Let $x \in Cl(A)$ and U be any open set containing x. If $A \cup U = \empty$ then $A \subset U^c$. Then Uc is a closed set containing A, so $Cl(A) \subset U^c$. This implies that $Cl(A)^c \subset U$ but $x \in U$, so $x \notin Cl(A)$, giving a contradiction. To prove the converse, suppose every open set U containing x intersects A but $x \notin Cl(A)$. That is, there is a closed set S containing A which does not contain x. Then, $x \in S^c \subset A^c$. But then Sc is an open set containing x, so it must have a non-empty intersection with A and therefore cannot be a subset of Ac, giving a contradiction.

The power of this theorem is made greater by noting that if the topological space in question has a basis, the condition on the sets U can be strengthened to not just open but also basic.

Scanned Notes

PDF

Available here (http://katlas.math.toronto.edu/0506-Topology/index.php?title=Image:2-1-09-22.pdf).

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