Examples of Continuous Functions in Finite Topological Spaces

Understanding continuous functions in the context of finite topological spaces is a fascinating topic in topology. A finite topological space is a set with a finite number of points equipped with a topology, which is a collection of open sets satisfying specific axioms. In this article, we will explore several examples of continuous functions between finite topological spaces, emphasizing the criteria for continuity.

Definition of a Finite Topological Space

A finite topological space is a fundamental concept in topology. It is defined as a set with a finite number of points that is equipped with a collection of open sets, known as a topology. The topology must meet the following axioms:

The empty set and the entire space are open sets. Arbitrary unions of open sets are open. Finite intersections of open sets are open.

Example 1: Identity Map in Discrete Topology

Let's consider the simplest example where both the domain and codomain are endowed with the discrete topology.

Consider the function f: {1, 2, 3, 4, 5} → {1, 2, 3, 4, 5} defined via the identity map. Both spaces are under the discrete topology, where every subset of the set is open.

Here, the preimage of any open set in {1, 2, 3, 4, 5} is the set itself, which is open. Therefore, the function is continuous.

Example 2: Identifying Finite Topological Spaces

For a more complex example, consider two finite topological spaces X and Y, where:

X {a, b} with the topology τ_X {?, {a}, {a, b}} Y {1, 2} with the topology σ_Y {?, {1}, {1, 2}}

A function f: X → Y is defined as:

f(a) 1 f(b) 1

To check the continuity of f, we need to verify that the preimage of every open set in Y is open in X.

Preimage of ?: f^(-1)(?) ?, which is open. Preimage of {1}: f^(-1)({1}) {a, b}, which is open. Preimage of {1, 2}: f^(-1)({1, 2}) {a, b}, which is open.

Since all preimages are open in X, the function f is continuous.

Non-Hausdorff Topologies and T0 Spaces

Finite topological spaces can be more interesting and varied than the simple examples given above. One interesting class of finite topological spaces are T0 spaces. In a T0 space, for each pair of distinct points, at least one of the points has an open neighborhood that does not contain the other.

A particularly useful concept for T0 spaces is de Groot duality, which establishes a relationship between the topology and a partially ordered set (poset). This duality allows us to investigate the continuity of functions in terms of the order of the poset.

Consider the following example:

(X, τ) ({a, b, c, d}, {?, {a}, {a, b}, {a, b, c, d}}) (Y, σ) ({1, 2, 3}, {?, {1}, {1, 2, 3}})

A function f: X → Y defined by:

f(a) 1 f(b) 1 f(c) 3 f(d) 3

To check the continuity, we examine the preimage of each open set in Y being open in X:

Preimage of ?: f^(-1)(?) ?, which is open. Preimage of {1}: f^(-1)({1}) {a, b}, which is open. Preimage of {1, 2, 3}: f^(-1)({1, 2, 3}) {a, b, c, d}, which is open.

The function f is continuous. Further, if the topologies on X and Y are derived from their respective order relations, the continuity of f can be checked by verifying a monotonicity condition: x ≤ x' implies f(x) ≤ f(x').

Thus, we have explored several examples of continuous functions in finite topological spaces and the criteria for their continuity, highlighting the importance of understanding both the topology and the order relations in these spaces.