Topological spaces, topological manifolds, and topological groups

Question

1. Topological space is defined more generally than topological manifolds. So all topological manifolds are topological spaces. But some topological spaces are not topological manifolds.

What are examples of topological spaces that are not topological manifolds?

2. Topological groups are logically the combination of groups and topological spaces, i.e. they are group and topological spaces at the same time, s.t. the continuity condition for the group operations connect these two structures together and consequently they are not independent from each other. So topological groups are also topological spaces.

What are examples of topological groups that are combination of groups and topological manifolds?

Lie groups are topological groups. Are finite groups also topological groups?

Answer 1

1. There are too many, you should take any topology textbook, find examples in that book and check whether they are manifolds. Take e.g. the Cantor set, indiscrete/discrete topologies, Zariski topologies, curves/surfaces/hypersurfaces with singularities, ...

2. Basic examples of topological groups that are also manifolds would be Lie groups - and the classical example of Lie groups are matrix groups like $\operatorname{GL}_n(\mathbb R), \operatorname{SL}_n(\mathbb R), O_n(\mathbb{R}), \operatorname{Sp}_n(\mathbb{R}), ...$. Finite groups can be made into topological groups by giving them the discrete topology (one often gives finite sets the discrete topology).

Btw, Lie groups are not only the basic examples of topological group which can be given a manifold structure, they are the only examples! This is answered by Hilbert's Fifth Problem.

Answer 2

$1.$ Another interesting example is the letters (as subset of $\mathbb{R}^2$).

$Y, X, A, H...$ all the letters that have segments with intersection outside of the vertices are not topological manifold. For example the letter $Y$:

If $Y$ is a topological $1-$manifold, then you can take a small open ball (we denote $\mathcal{B}$) that containg the center of the letter such that $\mathcal{B} \cap Y$ is homemorphic (we denote $\varphi$) to an open subset of $\mathbb{R}$. Since $\mathcal{B}\cap Y$ is connected, we have that $\varphi(\mathcal{B}\cap Y)$ is connected. Hence, as the connected of $\mathbb{R}$ are the intervals, we have that $\varphi(\mathcal{B}\cap Y)$ is an open interval.

If you remove the center, the letter $Y-\{center\}$ have three connected componentss but $\varphi(\mathcal{B}\cap Y)$ have two connected components. Since $\varphi: Y-\{center\}\to \varphi(\mathcal{B}\cap Y)-\{center\}$ still a homemorphism and homeomorphism preserves the number of connected components, we have a contradiction.

The same argument for the other letters is similar.

$2.1$ All finite group with discrete topology (all singleton are open subsets) is a $0-$dimensional topological manifold. For example the dihedral group $D_n$, the cyclic group $C_n$ and the permutation group $S_n$.

$2.2$ Finite group with discrete topology are also Lie Groups.