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 \((X,\mathcal{T})\). The collection \(\mathcal{T}_Y=\{O\cap Y:O\in\mathcal{T}\}\) is a topology on *Y* and called the subspace topology (aka relative topology, or induced topology, or topology induced on *Y* by \(\mathcal{T}\))

Note that an open set of \(\mathcal{T}\) is not necessarily an open set in \(\mathcal{T}_Y\) and vice versa. For example, the subspace \([0,1]\cup[3,4]\) of \(\mathbb{R}\) has some open sets such as \([0,1], [0,1),(0.5,1]\cup[3,3.5)\), etc.

A subspace *S* of \(\mathbb{R}\) is connected iff it is an interval, i.e. one of the following forms \(\{a\}, [a,b]\), \((a,b), [a,b)\), \((a,b], (-\infty, a)\), \((-\infty, a]\), \((a,\infty)\), \([a,\infty)\), \((-\infty,\infty)\)

A formal defintion of interval is a subset *S* in \(\mathbb{R}\) has the property: if \(x,z\in S,\ y\in\mathbb{R}\) and \(x\lt y\lt z\), then \(y\in S\).

## \(T_2\)-space (Hausdorff space)

A topological space \((X,\mathcal{T})\) is called a Hausdorff or \(T_2\)-space if given any pair of distinct points \(a,b\in X\), there exist open set \(U\ni a,\ V\ni b\) and \(U\cap V=\empty\).

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:

- \(\mathbb{R}\) is Hausdorff.
- Every discrete space is Hausdorff.
- Any Hausdorff space is also Fréchet (\(T_1\)-space).
- Any subspace of a Hausdorff space is Hausdorff.

## \(T_3\)-space (regular Hausdorff)

A topo space \((X,\mathcal{T})\) is called a regular space if for any closed \(A\subseteq X\) and \(x\notin A\), there exist an open set *U* containing *x*, an open set *V* containing *A*, such that \(U\cap V=\empty\).

If a regular space is also Hausdorff, we said it is a \(T_3\)-space or regular Hausdorff.

Some conclusions:

- Any subspace of a regular space is a regular space.
- \(\mathbb{R}, \mathbb{Q}, \mathbb{Z}, \mathbb{I}, \mathbb{R}^2\) are regular spaces.
- Any \(T_3\)-space is a \(T_2\)-space.

## Homeomorphisms

**Definition.** Let \((X,\mathcal{T})\) and \((Y,\mathcal{T}_1)\) be topo spaces. They are said to be homeomorphic if there exists a bijective function \(f:X\rightarrow Y\) which satisfies:

- For each \(U\in\mathcal{T}_1,\ f^{-1}(U)\in\mathcal{T}\)
- For each \(V\in\mathcal{T},\ f(V)\in\mathcal{T}_1\)

Furthermore, the map is said to be a homeomorphism between \((X,\mathcal{T})\) and \((Y,\mathcal{T}_1)\). We write \((X,\mathcal{T}) \cong(Y,\mathcal{T}_1)\).

\(\cong\) is an equivalance relation with reflexivity, symmetry, and transitivity.

Examples (\(a,b,c,d\in\mathbb{R}\)):

- Any two non-empty intervals (subspace of \(\mathbb{R}\)) are homeomorphic.
- If \(a<b,\ c<d\), then \([a,b]\cong[c,d]\).
- If \(a<b,\ c<d\), then \([a,b)\cong[c,d)\cong (a,b]\cong (c,d]\).
- If \(a<b,\ c<d\), then \((a,b)\ncong[c,d)\); \((a,b)\ncong [c,d]\); and \([a,b)\ncong [c,d]\).
- \((-\infty,a]\cong (-\infty,b]\cong [a,\infty)\cong [b,\infty)\).
- \(\mathbb{Z}\cong\mathbb{N}\)

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