Definition 14.13.
A property of a topological space \(X\) is a topological property (or topological invariant ) if every topological space homeomorphic to \(X\) has the same property.
Section Topological Invariants
Homeomorphic topological spaces are essentially the same from a topological perspective, and they share many properties, but not all. The properties they share are called topological invariants or topological properties.
Activity 14.6.
Which of the following are topological invariants? That is for topological spaces \((X, \tau_X)\) and \((Y, \tau_Y)\text{,}\) if \(X\) and \(Y\) are homeomorphic space and \(X\) has the property, does it follow that \(Y\) must also have that property?
(a)
\(X\) has the indiscrete topology
(b)
\(X\) has the discrete topology
(c)
\(X\) has the finite complement topology
(d)
\(X\) contains the number 2
(e)
\(X\) contains exactly 13 elements
