Is A Closed If Not Open in Topological Space?

[math8]

If A is not open in a topological space, does it follow A is closed?

 

[waht]

Not necessarily, the set A would be closed if X - A is open.

 

[math8]

If A, B are in a topology, does it imply A\B is in the Topology?

 

[waht]

If A, B are in a topology, does it imply A\B is in the Topology?

If A, B are in a topology then A and B are open. If I remember correctly, the definition of the topological space doesn't say anything about A\B.

 

[Tac-Tics]

If A is not open in a topological space, does it follow A is closed?

No. Topology is a little silly like that!

Here are the basic examples I use for each of the four possiblities. Each is written in standard interval notation.

open (but not closed)
(0, 1)

closed (but not open)
[0, 1]

neither closed nor open (I like to think of these as "half open" or "half closed")
[0, 1)
(0, 1]

both open and closed
$$\varnothing$$, $$\mathbb{R}$$

With these, you can think of more exotic examples in different spaces. In $$\mathbb{R}^n$$, for example, closed balls with a finite number of points removed will be neither open nor closed. In the set of integers using the discrete topology (or any set with the discrete topology), all points will be open AND closed.

Open AND closed sets are important when discussing connectedness. In a connected space (such as $$\mathbb{R}$$ or $$\mathbb{R}^n$$, the ONLY open-and-closed sets will be $$\varnothing$$ and the space itself.