Yes: two out of three axioms is enough.

No: finite intersections are required.

Yes, provided $X$ is finite.

No, because unions of any size are not allowed.

The discrete one.

The one a metric gives.

The indiscrete one.

They all have the same number.

$[0,1)$

$(0,1)$

$[0,1]$

$\mathbb{R}$

Any union of them is closed.

Any intersection of them is closed.

Any complement of them is closed.

None: closed sets combine in no reliable way.

$(0,1)$

$[0,1]$

$\mathbb{R}$

$\{0,1\}$

Every metric space is a topological space.

Every topological space comes from a metric.

A composite of continuous maps is continuous.

Every subset of a space is a space in its own right.

$N$ is open

Some open set $U$ satisfies $x\in U\subseteq N$

$x\in\partial N$

$N$ contains every open set around $x$

$\{7\}$

$(6,8)$

$\varnothing$

$\mathbb{R}$

Each of $(0,1)$, $[0,1]$ and $[0,1)$

Only $[0,1]$

Only $(0,1)$

None of them

$\mathbb{Z}$

$\mathbb{Q}$

$\{\,1/n:n\ge1\,\}$

$(0,1)$

Every subset is open

Any two non-empty open sets meet

The topology comes from a metric

The closed sets are the infinite sets

True

False