Connected space

Definition

A connected space is a topological space that cannot be divided into two disjoint non-empty open sets. This means that there is no way to separate the space into parts that do not touch each other, indicating a form of coherence within the space. Connectedness is essential when examining various properties of spaces, especially in relation to compactness and continuity.

5 Must Know Facts For Your Next Test

1. A connected space has the property that any continuous function from that space to the discrete topology must be constant.
2. If a connected space is also compact, it is particularly important in understanding various convergence behaviors and the nature of limit points.
3. The interval [a, b] on the real line is an example of a connected space, while the set of two separate points {1, 2} is an example of a disconnected space.
4. Connected spaces can be characterized using various properties like local connectedness, which relates to every point having a neighborhood base of connected sets.
5. In topological terms, the image of a connected space under a continuous function is also connected.

Review Questions

• What is the relationship between connected spaces and continuous functions, and why does this matter?
  • The relationship between connected spaces and continuous functions is significant because any continuous function from a connected space to a discrete topology must be constant. This highlights how connectedness impacts the behavior of functions, ensuring that once a point in the connected space is chosen, the image remains in one cohesive part rather than splitting into different values. Understanding this relationship helps clarify why connected spaces are vital in topology.
• How does compactness relate to connected spaces, and what implications does this have for their properties?
  • Compactness and connectedness are interrelated properties in topology. A compact connected space ensures that not only is it unable to be split into disjoint open sets, but it also possesses limits on how open covers behave. This means that if you cover the space with open sets, you can always find a finite number of them that still cover the whole space. This interdependence can be critical when studying convergence and continuity within such spaces.
• Evaluate how the concept of path-connectedness extends the idea of connectedness in topological spaces.
  • Path-connectedness extends the idea of connectedness by requiring that any two points in a space can be joined by a continuous path lying entirely within that space. While all path-connected spaces are connected, not all connected spaces are path-connected. This distinction becomes important in analyzing more complex spaces and understanding how different notions of connectivity can influence their topology, such as in applications involving homotopy or deformation.

Related terms

disconnected space: A disconnected space is one that can be separated into two or more disjoint non-empty open sets, implying that it lacks the property of connectedness.

path-connected space: A path-connected space is a stronger form of connectedness where any two points can be joined by a continuous path within the space.

compact space: A compact space is a topological space where every open cover has a finite subcover, often related to connectedness in terms of properties like being closed and bounded.