Intereting Posts

Paradox as to Measure of Countable Dense Subsets?
Prove that a continuous function on (0,1) has a sequence of step functions which converge pointwise to it on
Evaluating the log gamma integral $\int_{0}^{z} \log \Gamma (x) \, \mathrm dx$ in terms of the Hurwitz zeta function
Can only find 2 of the 4 groups of order 2014?
Why are the Trig functions defined by the counterclockwise path of a circle?
What are some counter-intuitive results in mathematics that involve only finite objects?
Why $\mathrm e^{\sqrt{27}\pi } $ is almost an integer?
Horse Race question: how to find the 3 fastest horses?
In the definition of a group, is stating the set together with the function on it redundant?
Are there spaces “smaller” than $c_0$ whose dual is $\ell^1$?
Isosceles triangle
Convert a Pair of Integers to a Integer, Optimally?
How to derive a function to approximate $\sqrt{3}$?
Exposed point of a compact convex set
how do i prove that $17^n-12^n-24^n+19^n \equiv 0 \pmod{35}$

In class we proved the following theorem:

Given $X_1,X_2$ ordered sets. Then any surjective increasing $\phi: X_1 \to X_2$ is continuous wrt the interval topology on $X_1$ and $X_2$.

I was asked to find an example to prove that the surjectivity condition above is essential: What does this mean? Do I need to find an increasing $f$ which is not surjective but continuous? How can I construct one?

- Equicontinuity on a compact metric space turns pointwise to uniform convergence
- Topological spaces in which every proper closed subset is compact
- Why this space is homeomorphic to the plane?
- Are the weak* and the sequential weak* closures the same?
- How many points does Stone-Čech compactification add?
- A bijective map that is not a homeomorphism

- Fundamental group of GL(n,C) is isomorphic to Z. How to learn to prove facts like this?
- Conditions for defining new metrics
- What is the smallest cardinality of a Kuratowski 14-set?
- Every path has a simple “subpath”
- General facts about locally Hausdorff spaces?
- Ref. Requst: Space of bounded Lipschitz functions is separable if the domain is separable.
- Finer topologies on a compact Hausdorff space
- The set of irrational numbers is not a $F_{\sigma}$ set.
- sum of two connected subset of $\mathbb{R}$
- How to prove that every uniform space is completely regular?

The problem is asking you to find a strictly increasing map $\phi : X_1 \rightarrow X_2$ that isn’t continuous, and to prove that $\phi$ isn’t continuous. The phrasing of the question offers an important hint, namely that you should be thinking about non-surjective choices of $\phi$, since no surjective choice is going to work.

Here’s a pretty substantial hint. Take $X_1$ equal to $\mathbb{N}$ with an element $a_0$ adjoined “at the end”, and $X_2$ equal to $\mathbb{N}$ with two elements $b_0$ and $b_1$ adjoined “at the end,” such that $b_0 \leq b_1$. Let $\phi : X_1 \rightarrow X_2$ be the map that takes $a_0$ to $b_1$, while leaving all the natural numbers unchanged. Now prove that $\phi$ isn’t continuous.

**Edit:** I now see that I misunderstood what you mean by increasing; you mean what I would call “strictly increasing.” I’ll leave this answer here anyway, since it shows that we cannot weaken this condition in the theorem.

**Original answer.** This isn’t true even if we assume surjectivity. Let $P = \{-1,1\}$ denote the ordered set such that $-1 \leq 1$. Then every subset of $P$ is open with respect to the interval topology. Now define $f : \mathbb{R} \rightarrow P$ by asserting that $f(x) = 1$ iff $x$ is positive. Then despite that $\{-1\} \subseteq P$ is open, nonetheless $f^{-1}(\{-1\})$ is not.

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- “If P, then Q; If P, then R; Therefore: If Q, then R.” Fallacy and Transitivity
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- Why is Lebesgue so often spelled “Lebesque”?
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- Derivative at $0$ of $\int_0^x \sin \frac{1}{t} dt$
- Factoring algebraic expressions of three variables
- Non-integer order derivative