# Connectedness and fineness of topology

In summary, if (X, T) is connected, then (X, T') is also connected, as T' is finer than T. The contrapositive is also true - if (X, T') is not connected, then (X, T) is not connected. This fact can be used to determine the connectedness of a space in a finer topology based on its connectedness in a coarser topology.
Homework Helper
Connectedness and "fineness" of topology

## Homework Statement

Let T and T' be two topologies on X, with T' finer than T. What does connectedness of X in one topology imply about connectedness in the other?

## The Attempt at a Solution

Assume (X, T) is connected, so there don't exist two disjoint, open and non-empty sets U, V whose union is X. Open sets U and V in T can be written as unions of basis elements Bx and By, where x are elements from U and y from V. Since T' is finer than T, U and V can as well be written as unions of basis elements B'x and B'y from T', where x and y are elements in U and V, respectively (since for any element x of U and Bx of T containing x there exists a basis element B'x of T' which contains x and is contained in Bx). So, connectedness of X in T implies connectedness of X in T', right?

Thanks for any replies, I hope I was clear enough, too lazy to TeX. :)

Btw, another question, of logical nature.

It is almost obvious that if (X, T) is not connected, neither is (X, T'), because of a similar argument, as the upper one. Does this mean that the negation of this statement is automatically true? It may be a stupid question, but I'm curious.

Any thoughts? I could make pertty much use of this fact, for example, I'm just solving the problem which asks to check whether R∞ is connected in the uniform topology. I know it's connected in the product topology, and since the uniform topology is finer than the product topology, if the above fact was true, I could conclude that it is connected in the uniform topology, too.

The reason for your confusion in the second post is that your argument in the first post proves the opposite of what you said it proves. Suppose $$(U, V)$$ disconnect $$(X, T)$$, that is $$U, V \in T$$ are a witness to the disconnectedness of $$(X, T)$$ because they are disjoint and their union is $$X$$. Because $$T'$$ is finer than $$T$$, that is $$T \subset T'$$, also $$U, V \in T'$$; therefore $$(U, V)$$ disconnect $$(X, T')$$. That is, if $$(X, T)$$ is not connected, neither is $$(X, T')$$. (Basis elements are irrelevant and not necessary to this argument.)

ystael said:
That is, if $$(X, T)$$ is not connected, neither is $$(X, T')$$. (Basis elements are irrelevant and not necessary to this argument.)

OK, this is clear to me now.

I'll quote a post from another thread I just found:

HallsofIvy said:
The "contrapositive" of the statement "if P then Q" is "if not Q then not P". Notice that we have not only changed each part to "not", we have swapped hypotheses and conclusion.

If the hypotheses cannot be true when the conclusion is false then knowing that the hypothesis is true tells us that the conclusion is true. A statement is true if and only if its contrapositive is true.

So, if (X, T') is connected, so is (X, T), right?

That's correct.

OK, thanks.

## 1. What is the difference between connectedness and fineness of topology?

Connectedness refers to the property of a topological space where every pair of points can be connected by a continuous path. Fineness of topology, on the other hand, refers to the amount of detail in a given topological space, with finer topologies having more open sets and thus more detail.

## 2. How are connectedness and fineness of topology related?

The two concepts are related in that a finer topology can potentially increase the connectedness of a space. This is because finer topologies have more open sets, which allows for more paths between points and thus a higher likelihood of connectedness.

## 3. How does the fineness of topology affect the continuity of functions?

Finer topologies can make it easier for functions to be continuous, as there are more open sets for the function to work with. However, in some cases, a function may be continuous with one topology but not with a finer topology.

## 4. Can a topological space be connected but not fine?

Yes, it is possible for a topological space to be connected but not fine. For example, a topological space with only one open set would be connected but not fine.

## 5. How does the concept of connectedness and fineness of topology apply to real-world situations?

In real-world situations, connectedness and fineness of topology can help determine the level of detail and continuity within a given space. For example, in a transportation network, a finer topology would allow for more efficient and connected routes between locations. In urban planning, the fineness of topology can impact the connectivity and accessibility of different neighborhoods.

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