Giving $\mathit{Top}(X,Y)$ an appropriate topology $\DeclareMathOperator\Top{\mathit{Top}}$I am not sure if its OK to ask this question here.
Let $\Top$ be the category of topological spaces. Let $X,Y$ be objects in $\Top$.
Let $F:\mathbb{I}\rightarrow \Top(X,Y)$ be a function (I will denote the image of $t$ by $F_t$). Let $F_{*}:X\times \mathbb{I}\rightarrow Y$ be the function that sends $(x,t)$ to $F_t(x)$.
Is there a topology on $\Top(X,Y)$ such that $F$ is continuous iff $F_{*}$ is continuous ?
Motivation: In the definition of a homotopy $F$ from $f$ to $g$ (for some $f,g\in \Top(X,Y)$) it is tempting to think of $F$ to be a "path" (as in the definition of $PY^X$) from $f$ to $g$. Now I really wanted to see if $F$ could be thought of as real path from $f$ to $g$ in $\Top(X,Y)$. More precisely, I wanted to know whether $F_{*}:\mathbb{I}\rightarrow \Top(X,Y)$ that sends $(x,t)$ to $F_t(x)$ is a path or not. Note that $F_{*}(0)=f,F_{*}(1)=g$,thus if $F_{*}$ is continuous it would be a path from $f$ to $g$ in $\Top(X,Y)$.
Hence, I still think that the case when $\mathbb{I}$ is the unit interval is still of some interest.
 A: It might be of interest to the original poster to know that $Top(X,Y)$ endowed with the compact-open topology guarantees at least one direction in the implication. In other words, if $F_*\colon X\times\mathbb{I}\to Y$ is continuous then $F\colon \mathbb{I}\to Top(X,Y)$ is. Hence you can safely interpret any homotopy as a path in the function space $Top(X,Y)$, but (allegedly) there are paths in $Top(X,Y)$ which do not correspond to homotopies.
I, for my part, would like to see a counterexample for the opposite direction, since all the counterexamples I know of seem to use a space different than $\mathbb{I}$.
Reference: Dugundji, Topology, Chapter XII, Theorem 3.1.
A: I guess you wish a "nice topology" for basic algebraic topology.
In the category of compactly generated spaces, the "compact open" topology behaves well, making this category a cartesian closed category.
Take a look at
https://en.wikipedia.org/wiki/Cartesian_closed_category
https://en.wikipedia.org/wiki/Compactly_generated_space
Your original question might be reformulated as: does the product functor $(\mathbb{I}\times -): Top\to Top $ have a right adjoint functor?
The answer is "no" and this is related with the bad behavior of the quotient topology with respect product topology, which implies bad behavior of the product with respect to pushouts.
https://en.wikipedia.org/wiki/Closed_monoidal_category , for aspects of category theory related with your question.
https://math.stackexchange.com/questions/31697/when-is-the-product-of-two-quotient-maps-a-quotient-map   (There are examples of products that don't preserve the quotient)
Here, there is a related question/answer
Categories with products that preserve quotients
Well, as I said, product doesn't preserve quotient (in the category Top) and, then, it doesn't preserve pushout. Therefore it's not left adjoint.
Also, you may argue that, if there is such a topology, you may conclude that products preserve quotient topology (which is an absurd).
Assume that there is such a topology: then, if $q:A\to B $ is a quotient map, take the product $q\times Z: A\times Z\to B\times Z $. We need to prove that, under our conditions, this map should be a quotient map (what is not true).
Given $f: B\times Z\to K $ a function, let
$(f\circ (q\times Z))_\ast : A\to Hom(Z,K) $
be the "adjoint" map of $(f\circ (q\times Z))$. By our hypothesis, $(f\circ (q\times Z))_\ast$ is continuous if, and only if,  $(f\circ (q\times Z))$ is so.
Note that
$(f\circ (q\times Z))_\ast $
is equal to
$f_\ast\circ q $
in which $f_\ast $ denotes the adjoint map of $f$. Since $q$ is quotient map,  $f_\ast\circ q$ is continuous if and only if $ f_\ast $ is continuous (which happens if and only if $f$ is continuous (by our hypothesis)).
Therefore $(f\circ (q\times Z))$ is continuous if and only if $f$ is continuous.
This would prove that $(q\times Z )$ is a quotient map, what isn't true in general.
And, therefore, we conclude that there isn't such a topology.
A: Briefly, this works very nicely when $X$ is locally compact, but not otherwise.
Then the function space carries the compact-open topology.
John Isbell gave a survey of the story and literature in  his paper
General Function Spaces, Products and Continuous Lattices,
in Math Proc Cam Phil Soc 100 (1986) 193--205.
It is an ongoing matter in theoretical computer science.
There is frequent and ongoing literature on this subject going back to
when Ralph Fox introduced the compact-open topology in
On Topologies for Function-Spaces in Bull AMS 51 (1945).
It was originally considered in homotopy theory,
then in category theory and topological lattice theory.
After that theoretical computer science took over,
under the headings of domain theory, realisability
and "exact" real computation.
Along the way some very important concepts have been identified,
in particular the universal property of the exponential in
a cartesian closed category (as stated elsewhere on this page)
but also that of a continuous lattice.
Briefly, a distributive continuous lattice is exactly the topology
of a locally compact space.
I say this primarily as a warning to those (students in particular)
who may think that a little bit of tweaking of the category or
the universal property might yield better results.
There are a lot of broken ideas along the way,
some of which you will find surveyed in Isbell's paper.
Breaking a correct idea like the universal property
(by restricting its test object to a single space) is not going to help.
The most important topological space is not the real interval but the
Sierpinski space, for which I write $\Sigma$.
Classically, it has open open and one closed point.
It is important because there are (constructive) bijections amongst

*

*continuous functions $\phi:X\to\Sigma$,

*open subspaces $U\subset X$ and

*closed subspaces $C\subset X$.

In particular, putting $Y\equiv\Sigma$ in the desired universal
property, a continuous map $\phi:\Gamma\times X\to\Sigma$
is an open subspace of $\Gamma\times X$
and you want that to correspond to a continuous function $\Gamma\to\Sigma^X$.
With $\Gamma\equiv{\bf 1}$, this means that the points of $\Sigma^X$
must be the open subspaces of $X$.
With $\Gamma\equiv\Sigma^X$, we want the transpose of $id:\Sigma^X\to\Sigma^X$
to be continuous, but this is $ev:\Sigma^X\times X\to\Sigma$ defined
by $ev(U,x)\equiv(x\in U)$.
This map defines an open subspace of $\Sigma^X\times X$,
which is a union of rectangles ${\cal V}\times V$.
If $x\in U$ then $(U,x)\in{\cal V}\times V$ and $x\in V\subset K\subset U$
where $K\equiv\bigcap{\cal V}$ is compact.
So this works exactly when $X$ is locally compact and $\Sigma^X$
is its lattice of open subspaces, itself equipped with the
Scott topology, which has a basis consisting of
${\cal V}\equiv\lbrace W|K\subset W\rbrace$ for $K$ compact.
I forget why $K$ is compact, but a good place to look would
be the paper
Local Compactness and Continuous Lattices
by Karl Hofmann and Mike Mislove
in Springer Lecture Notes in Mathematics 871 (1981) 209-248.
It was in this paper that the interpolation property
$x\in V\subset K\subset U$ was introduced
as the definition of a locally compact space that is (sober but)
not necessarily Hausdorff.
[PS:  Peter Johnstone has a neat argument involving preservation of injectivity, in the final chapter of his book Stone Spaces.]
So this is the reason why local compactness of $X$ is necessary.
If $X$ is locally compact then the exponentials $Y^X$ exist for all
spaces $Y$.
However, even when $Y$ is locally compact, $Y^X$ need not be,
for example Baire space $N^N$ is not,
so locally compact spaces do not form a cartesian closed category.
Nevertheless, $\Sigma^X$ is always locally compact when $X$ is.
Of course the argument for necessity above does not work
if you only allow $\Gamma\equiv[0,1]$ in the universal property.
However, it is not a good idea to mess around with such definitions.
If you seriously want to use the collection of maps $X\to Y$
as another space then you require a notation and a way of computing
with functions as first-class objects.
This notation is called the (typed) lambda calculus.
When the universal property of the exponential was recognised
in the 1960s, it was not only related to this question in general topology
but also to the formulation of symbolic logic,
that is, to the lambda calculus and to proof theory.
I always write $\Gamma$ for the test object of a universal property
because it plays exactly the same role in category theory
as the context does in symbolic logic,
which is customarily written with this letter.
The context of an expression is the list of parameters (free variables)
in it and their types (the spaces over which they range).
If you restrict  $\Gamma$ to be just the singleton or interval
then you cannot have general parameters in your expressions.
Dana Scott initially got involved in this subject because he wanted
to show that the untyped lambda calculus is meaningless.
However, he fairly quickly discovered models of it,
in the form of topological lattices such that $X\cong X^X$.
See, for example, his
Data Types as Lattices in
the SIAM Journal on Computing 5 (1976) 522-587.
Out of this grew veritable industries called domain theory
and denotational semantics.
In the 1980s, cartesian closed categories of domains came two-a-penny
(I was responsible for some of them), where
"domains" were particular kinds of partial orders
equipped with the Scott topology.
Denotational semantics used these to give mathematical
meanings to constructs in programming languages
in order to demonstrate the correctness of programs.
If you do not like the story for the whole of the traditional category
of topological spaces then there are many alternatives.
The "official" answer in homotopy theory was
the (full sub)category of compactly generated spaces.
On the other hand, there are ways of enlarging the traditional
category to make it cartesian closed.

Equilogical Spaces and Filter Spaces by Pino Rosolini
in Rendiconti del Circolo Matematico di Palermo 64 (2000) 157--175
gives an excellent survey of them,
explaining how they are reflective subcategories of
presheaves on the traditional category.
In particular, Scott had introduced equilogical spaces,
defined as topological spaces equipped with formal equivalence relations;
the theory is set out in full in

Equilogical Spaces
by Andrej Bauer, Lars Birkedal and Dana Scott.
Having gone to the trouble of writing this lengthy account
of (some of) the history of this question,
I would like to turn it back on the homotopy theorists.
When topics like this were considered by categorists in the 1960s,
they aimed their papers at (for example) topologists.
Therefore they did not spell out the topology, because their
intended readers would know it.
This is very frustrating for subsequent students of category theory:
the papers just contain the category theory and it is impossible
to trace back to the preceding mathematical ideas.
So I would be grateful if the homotopy theorists here would explain,
without rehearsing the category theory,
what the motivations were and are in their own subject
for asking for "convenient" or cartesian closed categories.
PS Thanks to Tyler Lawson for the comment below answering this question. Is there a slightly more detailed explanation of these methods, say of the length of a MO answer, or a survey paper?
In the context of an application of this kind, the next question is whether the cartesian closed categories that have been used (and mentioned above) are the most appropriate for the job.  On the face of it, you're happy with "any old" CCC.  But, when you look at the extra objects of this category, do the extensions of topological notions to them behave in the way that you would like? That is, according to whatever other intuitions of topology you have, such as developing results along the lines that Tyler mentions?
Many early applications of category theory imported the benefits of "set theory" by working in the Yoneda embedding (presheaves) or a smaller category of sheaves.  Rosolini showed (in the paper cited above) how the CCC extensions of categories of topological spaces are subcategories of the Yoneda embedding.  There is a close technical analogy in that both kinds of subcategory are reflective, but for sheaves the reflector (left adjoint to inclusion) preserves all finite limits, whereas in these CCCs it preserves products but not all equalisers or pullbacks.
My personal view is that these extensions are not topology but set theory with topological decoration. In this context, by "set theory" I mean, not the study of $\in$, but that of discrete spaces, whereas I believe (following Marshall Stone) that mathematical structures should be intrinsically topological. I have a research programme called Equideductive Topology that tries to look at such extensions without importing set theory.
A: The example you refer to (April 22) uses the idea of a placement of a body $B$ in an environment $E$, and notes that a path in the space of placements corresponds uniquely to a placement of $B$ in the space of paths, because both correspond to a lower-order map from $I\times B$ to $E$ itself. These correspondences are invertible, as well as smooth, recursive, etc. (i.e. they preserve whatever structure characterizes the ambient category of spaces). But the transformations are far from banal because it is the smoothness of functionals on the map spaces that the sought structure controls, and composing paths or placements with possible functionals yield new quantities of very diverse needed kinds.
This is described for example in Article V, Session 30, and Session 31 of Conceptual Mathematics (Lawvere & Schanuel). My paper about Volterra's functionals (published in the
RCM Palermo in 2000 and downloadable from my Buffalo homepage) discusses the unsuitability for functional analysis (as well as for homotopy theory) of the attempt to characterize continuity or cohesion using open sets or other contravariant structure.
That unsuitability was pointed out by Hurewicz who formulated the
necessary exponential laws and inspired Fox (BAMS 1945) to make use of the compact open construction that had been defined by Lefschetz in 1942. Hurewicz himself introduced, in Princeton lectures in the late 1940's, the k-spaces; these k-spaces seemed to be an adequate repair of the difficulty and so led to later studies with various refinements by Kelley, Steenrod, and (of course) yourself.
However, an equally serious flaw in the traditional definition of continuity had already been revealed by Peano's construction of space-filling curves, suggesting that a return to Frechet's covariant conception based on paths or on Eilenberg's singular figures should be considered. This is almost trivially the basis for the success of exponentials in simplicial sets and similar categories, including those studied in synthetic differential geometry; but the analysis given by Johnstone as a prelude to his important topological topos (which of course still contains the Peano pathologies) reveals that there are still significant foundational issues involved in the appropriate definition of a monoid of continuous reparameterizations of paths; these issues were emphasized by Grothendieck in his Tame Topology and have now been partially addressed by the o-minimal theorists.
(Hurewicz used the k to abbreviate 'kompakt erzeugte')
A: @Paul: Paul asks for the motivation: here is my story.
I gave  an MSc  course on  homotopy theory at Liverpool in 1960-61, and was struck then by the nice properties of the category of simplicial sets as against that of topological spaces, thus suggesting the convenience of simplicial sets. My thesis topic then was the algebraic topology of function spaces, and in the process of solving the particular problem I used exponential laws for spaces, simplicial sets, based simplicial stets, chain complexes, simplicial abelian groups, and maybe others. At the end of this work it struck me that the exponential law depended on the product as well as the hom, and I wrote this up as a small introduction.
I also knew that the weak (i.e. k-ified) product has been studied by Whitehead and by Danny Cohen, so it seemed reasonable to try this for the exponential law. To my surprise, it all worked well and became the first chapter of my thesis, on the category of Hausdorff k-spaces, which was submitted, and the thesis was reproduced  in the old purple Banda and well circulated, e.g. to Princeton.
Writing up this general topology part as papers it all became more expansive and was published as my first two papers, in 1963 and 1964.  In writing up the Introduction of the first paper, I speculated: "It may be that the category of Hausdorff k-spaces is adequate and  convenient for all purposes of topology." The major properties for convenience were listed in the second paper, mainly being cartesian closed.  I should say that a referee of the initial version had drawn my attention to the important point about cartesian closed, i.e. the usual properties of the product. Also, later workers eliminated the Hausdorff assumption.
For more discussion, see the the ncatlab on convenient categories of topological spaces.
There is also a nice paper of Lawvere discussing the various equivalences between $$(X^Y)^I, (X^I)^Y, X^{Y \times I}$$
in terms of motion and phase spaces, which I will try to find a reference to.
Finally, Spanier's suggestion of quasi-topological spaces is even more convenient, since it is locally cartesian closed, but was rejected mainly because the quasitopologies on the 2-point set formed a class. Maybe Peter Johnstone's "Topological topos" would be adequate and convenient for topology!
A: The answer is no in general as explained by the above answers. 
Since this problem is tagged algebraic topology, I guess that you care about function spaces because you care about homotopy theory. Here is how I think about function spaces when I am doing homotopy theory:
The space of functions from $X$ to $Y$ is the space representing the contravariant functor
$$ Z \mapsto {\rm maps}(X \times Z,Y) $$
this object might not exist in the category of topological spaces, but this does not really matter from the perspective of homotopy theory: you can just work inside a different category!
EDIT: 
I just want to add my opinion on Lennart's comment. Suppose that we "enlarge" the category of spaces to include the representing object ${\rm maps}(X,Y) $. We can extract a lot of information about the object ${\rm maps}(X,Y) $. Its points are just morphisms $ * \to {\rm maps}(X,Y)$ which we can identify with the set of maps from $X$ to $Y$. We also have a great description of maps from other spaces into ${\rm maps}(X,Y) $. I can't imagine that you can extract much more from the compact open topology (but I could be wrong here)
