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 $% $, then $[a,b]\cong[c,d]$.
• If $% $, then $[a,b)\cong[c,d)\cong (a,b]\cong (c,d]$.
• If $% $, 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