Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I would like some clarification on exactly what a 'continuous invariant' of topological spaces is. My book does not give a straight definition but rather just says "Properties preserved by continuous functions are called continuous invariants." It motivates this with the theorem that states "Let $X$ be a connected space and $f:X \to Y$ a continuous function from $X$ onto a space $Y$. Then $Y$ is connected."

In this case, the 'onto' part of the above theorem is necessary. For example, take a map $f:\mathbb{R} \to X : x \mapsto a$ where $a$ is a fixed point in $X$ and $X$ is a discrete space with at least 2 points. Then this constant function is continuous but $X$ is not connected, since no discrete space with at least two points is.

My question is: Is the 'onto' part of the theorem above part of the general definition of a continuous invariant? Does a continuous invariant carry over to the codomain of a continuous map that isn't surjective, or only to its image? What is the (or a) formal definition for "continuous invariant"?

share|improve this question

1 Answer 1

up vote 6 down vote accepted

Yes, the onto is understood. A property $P$ of topological spaces is a continuous invariant if whenever $X$ has $P$ and $f:X\to Y$ is a continuous surjection, then $Y$ has $P$. The idea is that $f$ carries the property from $X$ as a whole to $Y$ as a whole.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.