I made a table about topological spaces with or without connected, compact, and Hausdorff properties. However, I cannot find the example for the following cases:
Could you help me?
On
take $X=\mathbb R$ and define the topology $\tau=\{X,(-\infty,0),[o,\infty)\} $ it is easy to confirm that it is compact+not Hausdorff+not connected.
for the second case: take $Y=\prod _{\alpha\in\mathbb N}X_\alpha$ while $X_\alpha=X$ from the last case with the box topology.it is easy to see way it's not hausdorff+not connected. to show that it is not compact take the cover $$\{\prod_{\alpha<\beta}A_\alpha\times \prod_{\beta>0} B_\beta \}_\beta $$ while $A_\alpha=(-\infty,0) $ and $B_\alpha=[0,\infty)$
On
These are the simplest examples I could think of.
1) $X=\{1,2,3\}$ with topology $\{\varnothing,\{1\},\{2,3\},X\}$.
2) $X=\omega\cup \{\infty\}$ by declaring $\omega$ discrete and letting $\{0,\infty\}$ be the basic neighborhood of $\infty$.
On
$\pi$-Base is an online database of topological spaces inspired by Steen and Seebach's *Couterexamples in Topology.
This search result details three spaces that are compact, but neither Hausdorff nor connected:
This search result details two spaces that are non-compact, non-Hausdorff, and disconnected:
Take the disjoint union of the Sierpinski space with a point for 1. and infinitely many copies of that space for 2.
Edit: why your question (and my answer) is maybe not so interesting: as soon as you have a non-Hausdorff space, you also have a non-connected one (take disjoint union). As soon as you have a compact space, satisfying some "local-ish" property, you get a non-compact one, by taking infinitely many disjoint unions.
So actually the last part of my answer might be more interesting, since it tells you how to think in order to find counter-examples in topology.