DotKite said:Homework Statement
is it possible to have a topological space that is neither the indiscrete nor the discrete, and very set in the topology is clopen?
Homework Equations
The Attempt at a Solution
let ##X## = {(0,1),(2,3)} with the ordinary topology on R.
(0,1) is open, but it's complement which is (2,3) is open and which means (0,1) is closed. This (0,1) is clopen. Same argument for (2,3). Is this right?
A topological space is a mathematical concept that describes the properties of a set and the relationships between its elements. A space that is neither discrete nor indiscrete is one that does not have the properties of a discrete space (where every subset is open) or an indiscrete space (where only the empty set and the entire space are open).
A topological space that is neither discrete nor indiscrete is different from other types of spaces because it does not have the characteristics of a discrete or indiscrete space. It has a more complex structure and its open sets cannot be easily classified as either isolated or dense.
The main properties of a topological space that is neither discrete nor indiscrete include being non-discrete, non-indiscrete, and non-Hausdorff (meaning that there exist points that cannot be separated by disjoint open sets). It may also have other properties such as being compact or connected.
Topological spaces that are neither discrete nor indiscrete are used in scientific research to model complex systems and phenomena. They allow for a more nuanced understanding of the relationships between elements in a set and can help in the analysis and prediction of behavior in various fields such as physics, biology, and economics.
Yes, a topological space that is neither discrete nor indiscrete can be visualized, although it may be difficult to do so in some cases. It can be represented using diagrams and graphs that show the relationships between elements and open sets. It can also be visualized using computer simulations and other mathematical tools.