Is every subspace of a connected space connected?

[quantum123]

Is every subspace of a connected space connected?

 

[HallsofIvy]

What exactly do you mean by "space"- general topological space or topological vector space?

If you mean general topological space, the answer is obviously "no". Any subset of a topological space is a subspace with the inherited topology. A non-connected subset of a connected space with the inherited topology would be a non-connected space.

 

[quantum123]

Subspace I mean a subset with the induced subspace topology of a topological space (X,T).

 

[StatusX]

For a counterexample, take the real line, and the subset of the real line formed by removing a point.

 

[Hurkyl]

Or just take a pair of points. That's a disconnected subset too.

 

[mathwonk]

this is the most clueless question I've heard yet. reveals a complete lack of understanding of connectedness. presumably the questioner is learning by reading some worthless book with no examples or useful explanations at all.

(I waS IN exactly this boat when i was beginning topology, after hearing continuity defined as inverse image of opens are open. that is totally useless in understanding homeomorphisms. after that i still thought a sphere might be homeomorphic to a torus.)

 

[Hurkyl]

There are no stupid questions, only stupid answers.

 

[quantum123]

this is the most clueless question I've heard yet. reveals a complete lack of understanding of connectedness. presumably the questioner is learning by reading some worthless book with no examples or useful explanations at all.

(I waS IN exactly this boat when i was beginning topology, after hearing continuity defined as inverse image of opens are open. that is totally useless in understanding homeomorphisms. after that i still thought a sphere might be homeomorphic to a torus.)

Isn't that the definition of continuity? What is your definition? You mean without torus and sphere there would be no such thing as continuity?

 

[quantum123]

For a counterexample, take the real line, and the subset of the real line formed by removing a point.

Yes, the subset you mentioned is not connected with the usual or euclidean topology. But what about the induced or subspace topology where every open set is defined to be an intersection of that particular subset and a particular open set in the usual topology?

 

[matt grime]

You think it suddenly ceases to be disconnected? Take two open sets in the 'usual' topology that are disjoint and disconnect R\{0}. Each is open in the subspace topology by the very definition *you* wrote down.

Mathwonk's point was that sometimes it pays if you think about things. Whilst we always tell people that the first thing to do is check they know the definitions, the second thing is to see if they understand the definitions. This is different. This you do by playing with things and seeing what happens. It's good that you've learned the definitions, but the fact you had to make that last post asking that question indicates you need to think about the definitions some more for yourself, and in particular to actually use them and apply them to some questions.

 

[quantum123]

We are talking about topology and mathematics here. So is it better to stick to the topic and not talk about something else like for example, me?

 

[JasonRox]

Yes, the subset you mentioned is not connected with the usual or euclidean topology. But what about the induced or subspace topology where every open set is defined to be an intersection of that particular subset and a particular open set in the usual topology?

He gave a counter-example and which there are many!

Once you give a counter-example, the story of the question is done. If you want to ask a new one, go ahead, but that may require a different answer.

Oh, and by the way, he did use the subspace topology you're talking about.

 

[matt grime]

We are talking about topology and mathematics here.

We're talking about your understanding of topology.

So is it better to stick to the topic and not talk about something else like for example, me?

I can imagine it is not particularly pleasant, I apologise. Let me put it this way: there is a common philosophy amongst many mathematical educators that it is better to teach a man to fish than to give him a fish. It is more important to find out why, when given a counter-example, you didn't see it was a counter example and check the details rather than to give you more counter-examples.