## Munkres, Section 23 Connected Spaces

*A connected space*is one that cannot be separated into the union of two disjoint nonempty open sets. Otherwise such a pair of open sets is called*a separation of*.- Connectedness is a topological property: any two homeomorphic topological spaces are either both connected, or both disconnected, and the same set can be connected in one topology but disconnected in another, for example, and .
- A space is connected iff the only sets that are both open and closed in it are the whole space and the empty set.
- If a set is connected in a finer topology then it is connected in a coarser topology.

- Subspaces and connectedness:
- A subspace is disconnected iff there is a pair of disjoint nonempty subsets of whose union is , neither of which contains a limit point of the other (they may have a common limit point in but that does not count).
- A connected subspace of a disconnected space must lie entirely within a component of the separation.
- If any two sequential sets of a countable sequence of connected subspaces have a point in common then their union is connected.
- If a connected subspace have a common point with any set in a collection of connected subspaces then the union of the set with the union of the collection is connected.
- If a collection of connected subspaces have a point in common then their union is connected.

- If a subspace is connected then adding some of its limit points keeps it connected.
- This is useful, for example, for proving that the product space of any collection of connected spaces is connected in the product topology (see below).

- If and are connected and is a separation of then and are connected.

- Continuous functions and connectedness:
- The image of a connected space under a continuous function is connected.
- If is a quotient map, and are connected, then is connected.

- Products and connectedness.
- A finite product of connected spaces is connected.
- Proof for two spaces X and Y: for a fixed point y in Y: (X,y) is connected; for every x in X: (x,Y) is connected.

- An arbitrary product of connected spaces is connected in the product topology.
- Proof: fix a point in every space; a product of a finite number of spaces and fixed points in all other spaces is connected; the closure of the union of all such products is the whole space.
- is connected in the product topology but is not connected in the box or uniform topology.

- A finite product of connected spaces is connected.

## Munkres, Section 24 Connected Subspaces of the Real Line

*A linear continuum*is an ordered set such that the least upper bound property holds and for any pair of elements there is another one between them.- A subspace of a linear continuum is connected iff it is a convex subset.
- Any ordered set connected in the order topology is a linear continuum.
- If is well-ordered then is a linear continuum in the dictionary order.

*Generalized Intermediate Value Theorem*. If is a continuous function from a connected space to an ordered set in the order topology, and then there is such that .*Path connectedness*: given*a path from to*is a continuous function such that and . is said to be*path connected*if any two points in are path connected.- Path connectedness implies the connectedness. But not vice versa.
- The ordered square is connected but not path connected (the real line is just not enough for constructing the path).
*The topologist’s sine curve*is another example.- If a connected subspace of is open then it is path connected.

- (???) If a set is path connected and Hausdorff in one topology then it is not path connected in any strictly finer topology.

- Path connectedness implies the connectedness. But not vice versa.
- Subspaces and path connectedness:
- If a path connected subspace have a common point with any set in a collection of path connected subspaces then the union of the set with the union of the collection is path connected.
- If a collection of path connected subspaces have a point in common then their union is path connected.

- If a subspace is path connected then adding some of its limit points keeps it connected BUT it does not have to remain path connected.
- For example, the topologist’s sine curve is the closure of a path connected set.

- If a path connected subspace have a common point with any set in a collection of path connected subspaces then the union of the set with the union of the collection is path connected.
- Continuous functions and path connectedness:
- The image of a path connected space under a continuous function is path connected.

- Products and path connectedness.
- A finite product of path connected spaces is path connected.
- An arbitrary product of path connected spaces is path connected in the product topology.
- is path connected in the product topology but is not path connected in the box or uniform topology (it is not even connected in those two).

*The long line*: in the dictionary order with its smallest element deleted.- is path connected and locally homeomorphic to : is homeomorphic to an open interval in .
- cannot be embedded into or (it is not second-countable, see Exercise 7 of Section 30).

## Munkres, Section 25* Components and Local Connectedness

*A component of*is an equivalence class given by the equivalence relation: iff there is a connected subspace containing both.*A path component of*is an equivalence class given by the equivalence relation: iff there is a path connecting them.- Facts about (path) components:
- The (path) components of are (path) connected disjoint subspaces of whose union is such that each nonempty (path) connected subspace of intersects exactly one of them.
- Each path component lies within a component.
- Components are closed. If there are finitely many of them then they are open as well.

*Local (path) connectedness*. A space is*locally (path) connected at a point*if every neighborhood of the point contains a (path) connected sub-neighborhood. A space is*locally (path) connected*if it is locally (path) connected at every point.- A space is locally (path) connected iff every (path) component of every open set is open in .
- If a space is locally path connected then the components and the path components are the same.
- An open subset (as a subspace) of a locally (path) connected space is locally (path) connected.
- A connected open subset of a locally path connected space is path connected.

- Continuous functions and local connectedness:
- The image of a locally connected space under a quotient map is locally connected.

*A weakly locally connected at space*is a space such that every neighborhood of contains a connected subspace of which contains a neighborhood of . is weakly locally connected if it is weakly locally connected at every point.- Weak local connectedness of the whole space implies the local connectedness.
- Weak local connectedness at a point does NOT imply the local connectedness at the point.

*A quasi-component of*is an equivalence class given by the equivalence relation: iff there is no separation of into two open sets each containing a point or .- A component is contained within a quasi-component, so that if the space is connected then there is only one quasi-component.
- If a space is locally connected then the components and the quasi-components are the same.
- The quasi-component of containing is the intersection of all subsets of containing that are open and closed at the same time.

path connected | not path connected | ||||

connected | not connected | connected | not connected | ||

locally path connected (path components components) |
locally connected (components quasi-components) |
the indiscrete topology |
the discrete topology |
||

not locally connected | |||||

not locally path connected (path components components) |
locally connected (components quasi-components) |
the long circle (?) | |||

not locally connected (components quasi-components) |
the set of line segments: , | the topologist’s sine curve |

## Munkres, Section 26 Compact Spaces

*A compact space*is a space such that every open covering of contains a finite covering of .- If a space is compact in a finer topology then it is compact in a coarser one.
- If a space is compact in a finer topology and Hausdorff in a coarser one then the topologies are the same.
- Take a compact Hausdorff space. Any strictly coarser topology is not Hausdorff. Any strictly finer topology is not compact.
- For any set there is a topology such that it is a compact Hausdorff space.
- It is not true in general that for a compact non-Hausdorff topology there is a finer topology that is compact and Hausdorff; as well as, it is not true in general that for a non-compact Hausdorff topology there is a coarser topology that is compact and Hausdorff.

*The finite intersection property*: a collection of subsets is said to have the finite intersection property if every finite subcollection have a non-empty intersection.- A space is compact iff every collection of closed subsets having the finite intersection property has a nonempty intersection.
*Nested sequence*: if there is a nested sequence of closed nonempty subsets of a compact space then it has a nonempty intersection.

- A space is compact iff every collection of closed subsets having the finite intersection property has a nonempty intersection.
- Subspaces and compactness:
- A subspace is compact iff every covering by sets open in contains a finite sub-covering.
- A closed subspace of a compact space is compact.
- A compact subspace of need not be closed even if is compact: is not closed in (the topology is coarser than the standard topology, hence, since 1 is a limit point in the standard topology, it is a limit point in the given topology as well) but both are compact (the only open set containing 0 is the whole space). Another example is the cofinite topology on an infinite set: only finite subsets (and the whole space) are closed, but all subspaces (including the whole space) are compact.

- A compact subspace of a Hausdorff space is closed.
- Moreover, two compact subsets of a Hausdorff space can be separated by two open neighborhoods.

- A compact subspace of a metric space is closed and bounded.
- The finite union of compact subspaces is compact.
- The intersection of a nested sequence of connected closed subsets of a compact Hausdorff space is connected.
- We need the sequence to be nested: consider a unit square as a subspace of the standard plane and two disjoint points in it, and connect them by two different paths.

- Do we need (a) the subsets to be closed and (b) compactness for the non-emptiness of the intersection only? No. They are essential. We use the properties (a) and (b) in the proof several times: the intersection is closed (a) and, therefore, compact (b), a separation consists of two nonempty closed (a) and compact (b) sets (and can be separated by two disjoint neighborhoods), and every closed and connected subset in the sequence minus two disjoint neighborhoods of the separation is non-empty and closed (a), they are still nested, thus, their intersection is non-empty (b). Let f(x)=x(1-x). Then An=(0,0)U(1,0)U{0<y<f(x)/n|0<x<1} is a nested sequence of not closed but connected subsets of the compact and Hausdorff unit square with the disconnected intersection (0,0)U(1,0). Similarly, Bn=(0,0)U(1,0)U{y>=nf(x)|0<x<1} are connected, nested and closed in [0,1]x{0}U(0,1)xR+ (which is Hausdorff but not compact), but their intersection is the same discrete disconnected two-point set.
- What about the Hausdorff property? It seems that the only place we use the property in the proof is to find two disjoint open sets U and V in the space that each contains a set of a separation A and B of the intersection. What if we cannot find such neighborhoods? Consider the unit interval with double 0, it is compact and even connected (we don’t really need that), but not Hausdorff, though, it is T1. The sequence of subsets x<=1/n (including both 0’s) is a nested sequence of connected closed subsets with the intersection containing two 0’s only, a discrete disconnected subspace. The problem is exactly that the two 0’s have no disjoint neighborhoods.

- We need the sequence to be nested: consider a unit square as a subspace of the standard plane and two disjoint points in it, and connect them by two different paths.

- Continuous functions and compactness:
- The image of a compact set under a continuous function is compact.
- If compactHausdorff-space is continuous then it is a closed map.
- If compactHausdorff-space is bijective and continuous then it is a homeomorphism.

- If is continuous and closed then the preimage of any compact set is compact iff the preimage of any point is compact.
- If compactHausdorff-space is continuous then the preimage of any compact set is compact.

- where is compact and Hausdorff is continuous iff its graph in is closed.

*Perfect maps*: a closed continuous surjective function such that the preimage of every point is compact.- If is a perfect map and is compact, so is .

- Product and compactness:
- The product of finitely many compacts is compact.
- The product of any family of spaces is compact all of them are compact.
- is immediate, will be studied in §37.

*The tube lemma*:- Suppose and are arbitrary spaces and is compact, then if open in the product contains then it contains for some neighborhood of and of .
- If are compact subspaces of arbitrary spaces and an open set contains then contains a basis element that contains .

- If is compact then is a closed map.

- A partial converse to the uniform limit theorem:
- If a monotone increasing sequence of continuous functions on a compact domain converge (pointwise) to a continuous function then the convergence is uniform.

## Munkres, Section 27 Compact Subspaces of the Real Line

*Generalized Extreme Value Theorem*. If is a continuous function from a compact space to an ordered set in the order topology, then there are and : for all .- Ordered sets and compactness:
- A compact ordered set has the least and the largest elements.
- An ordered set satisfies the least upper bound property iff every closed interval is compact.
- An ordered set is compact iff it has the least and the largest elements and satisfies the least upper bound property.
- Consider : it is compact in the order topology, but not compact in the subspace topology.

- A subspace of is compact iff it is closed and bounded in the standard or square metric.
*Uniform continuity*: let and be metric spaces, then*The distance from a point to a set*is .*The Lebesgue Number Theorem*: If is compact and is a covering of then there is (*the Lebesgue number*) such that if the diameter of a set is less than then it is contained within a set in .*is uniformly continuous*if for every there is such that implies .*Uniform continuity theorem*: A continuous function from a compact metric space to a metric space is uniformly continuous.

- A nonempty compact Hausdorff space without isolated points is uncountable.
- For a countable sequence of points find a sequence of nested closed sets such that their intersection does not contain any points of the sequence.
- If is a countable collection of closed subsets of a compact Hausdorff space and each set has empty interior, then their union has empty interior as well.

- A connected metric space containing at least two points is uncountable.
*The Cantor set*: .- Compact, totally disconnected without isolated points and uncountable.

## Munkres, Section 28 Limit Point Compactness

*A limit point compact space (Bolzano-Weierstrass property, Fréchet compact, weakly countably compact)*is a space such that every its infinite subset has a limit point.*A sequentially compact space*is a space such that every sequence of points has a convergent subsequence.*A countably compact space*is a space such that every countable open covering has a finite subcovering.- A space is countably compact iff every countable nested sequence of closed nonempty subsets has a nonempty intersection.
- note that here the non-emptiness of the intersection of any nested collection of closed sets is not only necessary but sufficient as well

- A space is countably compact iff every countable nested sequence of closed nonempty subsets has a nonempty intersection.
- Relations:
- any space:
- compact or sequentially compact countably compact limit point compact
- take an infinite sequence of nested closed sets; if the space is compact then the intersection is not empty; if the space is sequentially compact then take a point in each set, there is a convergent subsequence and its limit must lie within the intersection
- any countable subset that contains no its own limit points can be covered by a countable number of open set containing one point of the subset only; there is no finite subcovering; if the space is countably compact, then the subset is not closed and has a limit point

- in general, there is no relation between compactness and sequential compactness.
- the first uncountable ordinal is not compact but it is sequentially compact
- the uncountable product of the unit intervals is compact but it is not sequentially compact

- compact or sequentially compact countably compact limit point compact
- a -space:
- limit point compact countably compact
- take an infinite sequence of nested closed sets and a point in each set; if the set of all points is finite, then the intersection is not empty, otherwise, there is a limit point; if the limit point is outside the intersection, then there is its neighborhood intersecting only finitely many sets of the collection; now we use the T
_{1}-property to construct another neighborhood that does not contain any points of the sequence different from the limit point

- take an infinite sequence of nested closed sets and a point in each set; if the set of all points is finite, then the intersection is not empty, otherwise, there is a limit point; if the limit point is outside the intersection, then there is its neighborhood intersecting only finitely many sets of the collection; now we use the T

- limit point compact countably compact
- a first-countable space (Section 30):
- countably compact sequentially compact
- take a sequence; suppose some point of the space is such that every its neighborhood contains a subsequence; now we use the first-countability: if there is a countable basis {B
_{n}} at the point, then for every n take a point of the sequence contained in B1∩…∩Bn with an index greater than the previous one — this subsequence has to converge to the point; now take the closure of the sequence; it is closed and, hence, countably compact; if all points in the closure have a neighborhood that does not contain any subsequence then there is no finite subcovering; therefore, some point in the closure is such that every its neighborhood contains a subsequence

- take a sequence; suppose some point of the space is such that every its neighborhood contains a subsequence; now we use the first-countability: if there is a countable basis {B

- countably compact sequentially compact
- a and first-countable space:
- limit point compact sequentially compact
- we may use the two previous statements but it is more useful to proof this directly: if a sequence contains a constant subsequence then the subsequence is convergent, otherwise there are infinitely many different elements in the sequence and it has a limit point; the T1 property guarantees that every neighborhood of the limit point contains a subsequence, and the first-countability ensures that we can construct a subsequence convergent to the limit point

- limit point compact sequentially compact
- a second countable space (Section 30):
- countably compact compact and sequentially compact (all three are equivalent)
- if there is a covering and a countable basis then take every basis open set that is contained in an open set of the covering and for each such basis set select one open set of the covering that contains it, this way we obtain a countable and then a finite subcovering
- if the space is second-countable then it is first-countable

- countably compact compact and sequentially compact (all three are equivalent)
- a metric space or a second-countable -space:
- all four are equivalent
- a metric space is first-countable and T1, therefore, (c implies cc, lpc and sc) and (cc iff lpc iff sc)
- now, if we knew that the space is also second countable, then we would conclude that all four are equivalent, however, there are metric spaces that are not second-countable (for example, discrete uncountable space or a fancier one: take the unit square and suppose that one can move horizontally at the bottom of the square only), but…

- if a metric space is limit point compact then it is second-countable
- first, for every positive r there is a covering by a finite number of r-balls (if there is no finite covering by r-balls then there is an infinite set of points such that the distance between any two points is at least r but it has no limit points); second, take the union of all finite collections of 1/n-balls covering the space — it is a countable basis

- all four are equivalent to the requirement that the space is bounded under every metric that induces the topology
- see Exercise 3 of Section 35: if compact then the distance being continuous is bounded; if every metric is bounded and there is a continuous function then we can construct a metric such that it is bounded iff the function is bounded
- idea: this, actually, provides another way to show that countable compactness of a metric space implies compactness, namely, if countably compact then must be bounded in any metric, therefore, compact

- all four are equivalent

- any space:
- Examples:
- is not compact (no largest element) but it is limit point compact (an infinite set has a countable subset which is bounded and lies within a compact closed interval).
- is compact and, therefore, limit point compact.

- Subsets and limit point compactness:
- A closed subset of a limit point compact space is limit point compact.
- A limit point compact subset of a Hausdorff space does NOT have to be closed (even if the space is compact).
- For example, .

- Continuous functions and limit point compactness:
- The image of a limit point compact set under a continuous function does NOT need to be limit point compact.

- Products and limit point compactness:
- Even the product of two Hausdorff limit point compact spaces does NOT have to be limit point compact.

*An isometry*() from a compact metric space into itself is a homeomorphism (i.e. it must be surjective).*A shrinking map*() from a compact metric space into itself has a unique*fixed point*: .*A contraction*( for some ) from a compact metric space into itself has a unique fixed point (this result for contractions only, i.e. not for shrinking maps in general, can be generalized to non-compact complete metric spaces, see Section 43).

## Munkres, Section 29 Local Compactness

*A locally compact at a point space*is a space that contains a compact subspace containing a neighborhood of the point.*A locally compact space*is a space that is locally compact at each of its points.- A Hausdorff space is locally compact iff any neighborhood of any point contains a compact closure of a neighborhood of the point.

Two of the most well-behaved classes of spaces to deal with in mathematics are the metrizable spaces and the compact Hausdorff spaces… If a given space is not of one of these types, the next best thing one can hope for is that it is a subspace of one of these spaces… Thus arises the question: Under what conditions is a space homeomorphic with a subspace of a compact Hausdorff space?

*One-point compactification*of a locally compact Hausdorff space:- is locally compact Hausdorff iff is a subspace of a compact Hausdorff space .
- If is locally compact Hausdorff then the one-point compactification is unique up to a homeomorphism that preserves points of .
- One-point compactification seems to work for any Hausdorff space but is Hausdorff for locally compact spaces only.

- Subspaces and local compactness:
- A closed subspace of a locally compact space is locally compact.
- An open subspace of a locally compact Hausdorff space is locally compact (and Hausdorff).
- is locally compact Hausdorff iff it is homeomorphic to an open subspace of a compact Hausdorff space.

- Continuous functions and local compactness:
- The open continuous image of a locally compact space is locally compact.
- In general, the continuous image of a locally compact space does NOT have to be locally compact.
- A homeomorphism of two locally compact Hausdorff spaces can be extended to a homeomorphism of their one-point compactifications.

- Quotient maps and local compactness:
- If is a quotient map and is locally compact and Hausdorff then is a quotient map.
- If are quotient maps and (or ) are locally compact and Hausdorff then is a quotient map.

- Products and local compactness:
- The finite product of locally compact spaces is locally compact.
- The product of any family of spaces is locally compact all but finitely many of them are compact and those which are not compact are locally compact.

- Topological groups and local compactness:
- If a topological group is locally compact then any its quotient group is locally compact as well.

*^{(m)} means metrizable |
countably compact | not countably compact | ||||

limit point compact | not limit point compact | limit point compact | not limit point compact | |||

sequentially compact | compact | ^{(m)} |
||||

not compact | locally compact | |||||

not locally compact | ||||||

not sequentially compact | compact | |||||

not compact | locally compact | ^{(m)}^{(m)} |
||||

not locally compact | ^{(m)}^{(m)} |

## Supplementary Exercises*: Nets

*A directed set*is a partially (strictly) ordered set such that for any pair of elements there is such that and .- We can require it to be a strict partial order or a weak partial order: the definitions and results below hold in either case

*A cofinal subset*of a partially ordered set is a subset such that for any element in there is a greater element in .- If is a directed set then its cofinal subsets are directed as well.

*A net in a topological space*is a function from a directed set into : where .- In other words, a net is a subset of the space indexed by a directed set.
- A sequence is a net.

*A subnet of the net*is where is such that and is cofinal in .- A subsequence is a subnet, but not every subnet of a sequence is a subsequence.
- In fact, if a subnet was defined as a cofinal subset of a net then every subnet would be a subsequence, but then this definition would not be “flexible” enough for some criteria stated below. For example, a sequence may have a convergent subnet but no convergent subsequence.

- A subsequence is a subnet, but not every subnet of a sequence is a subsequence.
*The net converges to a point*if for each neighborhood of there is such that implies . We denote this by .- If is Hausdorff then a net cannot converge to more than one point.
- If , then .
- A net converges to iff every its subnet converges to .

*An accumulation point of net*is a point such that for every its neighborhood the subset of indexes such that is cofinal in .- A net has an accumulation point iff it has a subnet converging to .

- The closure and nets.
- iff there is a net of points of converging to .
- Compare to the Sequence Lemma of Section 21:
- if there is a sequence of points of converging to then the sequence is a net in and
- but if then there is just a net in converging to and this does not guarantee the existence of a sequence of points of converging to : something we can show if the space is metrizable

- Compare to the Sequence Lemma of Section 21:

- iff there is a net of points of converging to .
- Continuous functions and nets.
- is continuous iff for every convergent net : .
- Compare to Section 21:
- if is continuous then the image of any net converging to any is a net (indexed by the same directed set) converging to , in particular, this is true for any convergent sequence of points of
- but if the image of any sequence converging to any converges to , this can still be not enough for the continuity of : it is enough if the space is metrizable

- Compare to Section 21:

- is continuous iff for every convergent net : .
- Compactness and nets.
- is compact iff every net in has a convergent subnet.
- Compare to Section 28:
- if a space is compact then every sequence has a convergent subnet, but this does NOT imply that the sequence has a convergent subsequence
- also if every sequence has a convergent subsequence this implies that every sequence has a convergent subnet but does not imply that every net has a convergent subnet
- given these two, compactness and sequential compactness are not comparable

- Let and be the nth digit of the binary expansion of (no tails of 1’s). is compact but not sequentially compact.
- For any subsequence let be such that its ’s digit of the binary expansion is . Then does not converge.
- Let , . Then is not empty. Every point in is an accumulation point of the sequence. And vice versa. One cannot give an explicit example of such a point.

- Compare to Section 28:

- is compact iff every net in has a convergent subnet.
- Topological groups (an application):
- If is a closed subset and is a compact subset of a topological group then is closed in .

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