A summary of Chapter 4 and 5 of the book "Topology without Tears" by Sidney Morris that I'm reading.
Subspace
Definition. Let Y be a non-empty subset of the topo space . The collection is a topology on Y and called the subspace topology (aka relative topology, or induced topology, or topology induced on Y by )
Note that an open set of is not necessarily an open set in and vice versa. For example, the subspace of has some open sets such as , etc.
A subspace S of is connected iff it is an interval, i.e. one of the following forms , , , , , ,
A formal defintion of interval is a subset S in has the property: if and , then .
-space (Hausdorff space)
A topological space is called a Hausdorff or -space if given any pair of distinct points , there exist open set and .
In other words, for any two distinct points there exists a neighbourhood of each which is disjoint from the neighbourhood of the other.
Some conclusions:
- is Hausdorff.
- Every discrete space is Hausdorff.
- Any Hausdorff space is also Fréchet (-space).
- Any subspace of a Hausdorff space is Hausdorff.
-space (regular Hausdorff)
A topo space is called a regular space if for any closed and , there exist an open set U containing x, an open set V containing A, such that .
If a regular space is also Hausdorff, we said it is a -space or regular Hausdorff.
Some conclusions:
- Any subspace of a regular space is a regular space.
- are regular spaces.
- Any -space is a -space.
Homeomorphisms
Definition. Let and be topo spaces. They are said to be homeomorphic if there exists a bijective function which satisfies:
- For each
- For each
Furthermore, the map is said to be a homeomorphism between and . We write .
is an equivalance relation with reflexivity, symmetry, and transitivity.
Examples ():
- Any two non-empty intervals (subspace of ) are homeomorphic.
- If , then .
- If , then .
- If , then ; ; and .
- .
Properties preserved by homeomorphisms:
- T0-space, T1-space, T2-space, regular space, T3-space
- second countable
- separable space
- discrete, indiscrete, finite-closed, countable-closed
- cardinality