En anglais

Continuity with open sets

Mathématiques · Topologie · de 23 à 24 ans

Nom ______________________ Date ____________

How many open sets does the discrete topology on a 3-point set have?
Réponse : ______________
Which collection is a topology on {1, 2}?
- {empty set, {1}, {2}}
- {{1}, {1, 2}}
- {{1}, {2}, {1, 2}}
- {empty set, {1}, {1, 2}}
How is continuity defined with open sets?
- The preimage of every open set is open
- The image of every open set is open
- The preimage of every closed set is open
- The image of every closed set is closed
Arbitrary unions of open sets are always open.
En entourer un : Vrai Faux
How many open sets does the indiscrete topology on a 4-point set have?
Réponse : ______________
How do the epsilon-delta and preimage definitions of continuity compare?
- They disagree on the real line
- The preimage version is strictly weaker
- They agree on every metric space
- Epsilon-delta applies to spaces without distance
A student says every function out of a discrete space is continuous. Is that right?
En entourer un : Vrai Faux
The identity map from a discrete space to an indiscrete one is continuous, but the reverse is not.
En entourer un : Vrai Faux
A proposed collection holds the whole set and closes under unions but skips the empty set. What follows?
- It still qualifies through the union axiom
- It is not a topology: the empty set is mandatory
- It qualifies on any two-point set
Take one set with two comparable topologies. In which direction is the identity map always continuous?
- From the coarser space to the finer one
- From the finer space to the coarser one
- In neither direction
- In both directions always

Corrigé

Pour les adultes. Replier cette page avant de passer le reste.

Continuity with open sets W1-mt_D2oPq0AVhI-s1

8 · Every subset is open in the discrete topology, and a 3-point set has 2^3 = 8 subsets.
{empty set, {1}, {1, 2}} · The axioms demand the empty set, the whole set, and closure under unions; only {empty set, {1}, {1, 2}} meets all three.
The preimage of every open set is open · Continuity is defined by open preimages: the preimage of every open set is open.
Vrai · Closure under arbitrary unions is the topology axiom.
2 · The indiscrete topology has only the empty set and the whole 4-point set.
They agree on every metric space · Open balls bridge the two: they agree on every metric space.
Vrai · Every subset of a discrete domain is open, so every preimage is open.
Vrai · The topology choice decides, and the two directions differ.
It is not a topology: the empty set is mandatory · Missing the empty set fails the axioms at once.
From the finer space to the coarser one · Coarsening the codomain only removes checks, so from the finer space to the coarser one is always continuous.