Open/closed function [closed]

1. Can a closed set be open?
2. Are all open maps closed?
3. What makes a function closed?
4. How do you prove a map is closed?
5. Is R closed?
6. Is zero set closed?
7. Are continuous functions closed?
8. Is sin a closed function?
9. Is the image of a closed set closed?
10. Can a function be closed?
11. Is QA closed set?
12. How do you tell if a function is open or closed?

Can a closed set be open?

Sets can be open, closed, both, or neither. (A set that is both open and closed is sometimes called "clopen.") The definition of "closed" involves some amount of "opposite-ness," in that the complement of a set is kind of its "opposite," but closed and open themselves are not opposites.

Are all open maps closed?

Likewise, a closed map is a function that maps closed sets to closed sets. ... A map may be open, closed, both, or neither; in particular, an open map need not be closed and vice versa.

What makes a function closed?

A closure is the combination of a function bundled together (enclosed) with references to its surrounding state (the lexical environment). In other words, a closure gives you access to an outer function's scope from an inner function.

How do you prove a map is closed?

The closed map lemma says that if f:X→Y is a continuous function, X is compact and Y is Hausdorff, then f is a closed map.

Is R closed?

The empty set ∅ and R are both open and closed; they're the only such sets. Most subsets of R are neither open nor closed (so, unlike doors, “not open” doesn't mean “closed” and “not closed” doesn't mean “open”).

Is zero set closed?

So the only boundary point of [0,∞) and (0,∞) is 0 itself. It is in [0,∞), so that set is closed.

Are continuous functions closed?

A function f : X → Y is called continuous if the preimage under f of any open subset of Y is an open subset of X. ... f is continuous if and only if the preimages under f of closed subsets are closed.

Is sin a closed function?

A continuous map which is not open nor closed

It is well known that sin is continuous. sin is not open as the image of the open interval (0,π) is the interval (0,1].

Is the image of a closed set closed?

If instead we were dealing with closed and bounded sets, then their images would always be closed (and bounded). This is a result that can be summarized by saying "the image of a compact set under a continuous function is compact".

Can a function be closed?

A proper convex function is closed if and only if it is lower semi-continuous. For a convex function which is not proper there is disagreement as to the definition of the closure of the function.

Is QA closed set?

In the usual topology of R, Q is neither open nor closed. The interior of Q is empty (any nonempty interval contains irrationals, so no nonempty open set can be contained in Q). Since Q does not equal its interior, Q is not open. ... Since Q does not equal its closure, it is not closed.

How do you tell if a function is open or closed?

A domain (denoted by region R) is said to be closed if the region R contains all boundary points. If the region R does not contain any boundary points, then the Domain is said to be open. If the region R contains some but not all of the boundary points, then the Domain is said to be both open and closed.