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

Reference sources

  1. Sidney Morris - “Topology Without Tears”