Topology 12-17

Topology Definition

1) Empty set and X are in T. 2) The Union of the elements of any subcollectiom of T is in T. 3) The intersection of the elements of any finite subcollection of T is in T.

Hausdorff Space

A space X is Hausdorff if for every pair of distinct points x1, x2 there exist open sets U1, U2 such that x1 E U1, x2 E U2, U1 ∩ U2 = Ø

Subbasis for a Topology

A subbasis S for a topology on X is a collection of subsets of X whose union equals X. The topology generated by the subbasis is defined to be the collection T of all unions of finite intersections of element of S.

Subbasis

A subbasis for a topology on X is a collection S of subsets of X, whose union equals X. The basis associated to S is the collection B consisting of all finite intersections B = S1 ∩ · · · ∩ Sn of elements S1, . . . , Sn ∈ S , for n ≥ 1. By the topology T generated by the subbasis S we mean the topology generated by the associated basis B. Clearly S ⊂ B ⊂ T .

Open Set

A topological space is a pair (X, T ) where X is a set and T is a family of subsets of X (called the topology of X) whose elements are called open. If X is a topological space with topology T, we say that a subset U of X is an open set of X if U belongs to the collection T. In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line.

Properties of Closed Sets

Any intersection of closed sets is closed (including intersections of infinitely many closed sets) The union of finitely many closed sets is closed. The empty set is closed. The whole set is closed.

Let X and Y be topological spaces. Describe a basis for the product topology on X × Y .

B = {U × V |U is open in X and V is open in Y }

Example of a Topology on R (lower limit topology)

Consider the collection B of subsets in R: B :=([a, b) := {x ∈ R | a ≤ x < b}|a, b ∈ R)

Subspace

Definition. Let X be a topological space with topology T . If Y is a subset of X, the collection TY = {Y ∩ U | U ∈ T } is a topology on Y , called the subspace topology. With this topology, Y is called a subspace of X.

A sub set that is both open and closed

Example Let X = R and Y = [0, 1]∪(2, 3). We can endow Y with the subspace topology coming from X. The subset (2,3) ⊂ Y is open in Y , as (2, 3) = (2, 3) ∩ Y , and (2, 3) is openas a subset of R. Similarly, [0, 1] is open as a subset of Y , for [0, 1] = (−1, 2)∩Y , and (−1, 2) is open in R. Thus [0, 1] and (2, 3) are both open and closed. This is a good example of a disconnected space - one in which there are non-trivial (not X or ∅) subsets which are both closed and open.

Examples of a topology on R other than the standard topology

For example, R = {(a, ∞) : a ∈ R}∪{R, ∅} is a topology on R called the right ray topology. The collection F = {S ⊂ R : Sc is finite} ∪ {∅} is called the finite complement topology.

Closure and Interior of a Set

Given a subset A of a topological space X, the interior of A is defined as the union of all open sets contained in A, and the closure of A is defined as the intersection of all closed sets containing A. Notation. The interior of A is denoted Int A and the closure of A is denoted by Cl A or A. Remark. Obviously, we have Int A ⊂ A ⊂ A.

Basis Elements

If X is a set, a basis for a topology on X is a collection B of subsets of X such that: 1) for each xEX,there is at least one basis element B containing x. 2) If x belongs to the intersection of two basis elements B1 and B2, then there is a basis element B3 containing x such that B3 ⊂ B1 ∩ B2.

Closed Set

In a topological space, a closed set can be defined as a set which contains all its limit points.

Topological Space

Is an ordered pair (X,T) consisting of a set X and a topology T on X.

Standard Topology

Let R be the set of all real numbers. Let B be the collection of all open intervals: (a, b) := {x ∈ R | a < x < b}.

a topological space X with subspace Y and subset A ⊆ Y so that A is open in Y , but A is not open in X.

Let X = R, and let Y = [0, 2]. Then the set A = [0, 1) is open in Y as A = (−1, 1) ∩ Y and (−1, 1) is open in X.

a subset of a topological space that is both open and closed.

Let X = {a, b, c} in the discrete topology T = {∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, X}. Every set subset of X is both open and closed in this topology.

A subset of a topological space that is neither open nor closed

Let X be a set. The finite complement topology has all its open sets as well as the set ∅. Thus the complements of the open sets are the finite sets, as well as the total set X. So the closed sets in the finite complement topology are the finite subsets of X as well as X. Notice that in the integers, equipped with the finite complement topology, the collection of even integers is neither open nor closed.

Brandl definition of limit point

Let X be a topological space, A⊂X, xEX is a limit point of A if every neighborhood of x intersected point other than x. x E X is a limit point of A if every neighborhood of x intersects A in a point other than x.

Product Topology

Let be X and Y be topological spaces. The product topology on X x Y is the topology having as basis the collection B of all sets of the form U x V, where U is an open subset of X and V is an open subset of Y.

a topological space X with subspace Y and subset A ⊆ Y so that A is open in X, but A is not open in Y

None exists: a set U is open in Y if U = V ∩ Y where V is open in X. This means that if A ⊆ Y and A is open in X, then A = A ∩ Y is open in Y .

6. Prove that the closure of a set A consists of the set A together with the set of limit points of A.

Proof. Let A0 be the set of limit points of A. We will show that A¯ ⊆ A ∪ A0 and that A ∪ A0 ⊆ A¯. Let x ∈ A¯. If x ∈ A, then certainly x ∈ A ∪ A0 , so assume that x 6∈ A. Since x ∈ A¯, by a previous theorem, every open set U containing x intersects A. Since x is not in A, such a U must intersect A at a point other than x, so x ∈ A0 . This completes the proof that A¯ ⊆ A ∪ A0 . It remains to show that A ∪ A0 ⊆ A¯. Of course A ⊆ A¯, so we only need to show that A0 ⊆ A¯. So let x ∈ A0 . Then every open neighborhood of x intersects A at a point different from x, so by previous theorem (the statement proved in problem 5 above),

Let A be a subset of the topological space X. Prove that x ∈ A¯ if and only if every open set U containing x intersects A.

Proof. We will prove the contrapositive of the statement which is: x 6∈ A¯ if and only if there exists an open set U containing x that does not intersect A. So suppose that x 6∈ A¯. Then the set U −A¯ is open and contains x and does not intersect A. On the other hand, suppose there is a set U that contains x and does not intersect A. Then X − U is a closed set that contains A. By the definition of closure, A¯ ⊆ X − U. Since x ∈ U, we can conclude that x 6∈ A¯.

Let {Tα} be a collection of topologies on a set X. Is ∪α Tα a topology on X? Why or why not? Describe the biggest topology T such that T ⊆ Tα for each α. Describe the smallest topology Ts such that Tα ⊆ T for all α.

The collection ∪α Tα is not necessarily a topology on X. For example, if U is in Tα and V is in a different topology Tβ, then U ∩ V is not necessarily in ∪ α Tα. However, T = ∩ α Tα is a topology on X and is the largest topology contained in each Tα. To prove this we need to show that ∅, X ∈ T , and that T contains finite intersections and arbitrary unions of elements of T . Well, each Tα is a topology, so ∅, X ∈ Tα for each α, therefore ∅, X ∈ ∩ α T = T . Let Ui ∈ T , for i = 1, 2, . . . , n. Then for each α, Ui ∈ Tα for each i = 1, 2, . . . , n. Since Tα is a topology, this means that n ∩ i=1 Ui ∈ Tα for each α. But then n ∩ i=1 Ui ∈ ∩ α T = T . Similarly, if {Uβ}β∈J is a collection of sets in T , then for each α, Uβ ∈ Tα, so ∪ β∈J Uβ ∈ Tα for all α (since each Tα is topology). Then ∪ β∈J Uβ ∈ ∩ α Tα = T . This completes the proof that T = ∩α Tα is a topology on X. Since T is the largest set that is contained in each Tα, it must be the largest topology such that T ⊆ Tα for each α. The smallest topology Ts such that Tα ⊆ Ts for all α is the topology generated by the subbasis S = ∪ α Tα. Then the sets in Ts are unions of finite intersections of sets in ∪ α Tα. Certainly this topology contains each Tα, and any topology that contains each Tα must also contain Ts.

a subset of a topological space that is neither open nor closed.

The subset [0, 1) of R in the standard topology is neither open nor closed.

Limit Point of a Set

a limit point of a set S in a topological space X is a point x (which is in X, but not necessarily in S) that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contains a point of S other than x itself. Note that x does not have to be an element of S. This concept profitably generalizes the notion of a limit and is the underpinning of concepts such as closed set and topological closure. Indeed, a set is closed if and only if it contains all of its limit points, and the topological closure operation can be thought of as an operation that enriches a set by adding its limit points. Let S be a subset of a topological space X. A point x in X is a limit point of S if every neighbourhood of x contains at least one point of S different from x itself. Note that it doesn't make a difference if we restrict the condition to open neighbourhoods only.

A neighborhood

is of x E X is an open set that contains x.

Closure of a Set

the closure of a subset S in a topological space consists of all points in S plus the limit points of S. The closure of S is also defined as the union of S and its boundary. Intuitively, these are all the points in S and "near" S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior.