munkres topology homework solutions

Munkres (2000) Topology with Solutions

Below are links to answers and solutions for exercises in the Munkres (2000) Topology, Second Edition .

• Section 1: Fundamental Concepts
• Section 2: Functions
• Section 3: Relations
• Section 4: The Integers and the Real Numbers
• Section 5: Cartesian Products
• Section 6: Finite Sets
• Section 7: Countable and Uncountable Sets
• Section 8*: The Principle of Recursive Definition
• Section 9: Infinite Sets and the Axiom of Choice
• Section 10: Well-Ordered Sets
• Section 11*: The Maximum Principle
• Supplementary Exercises*: Well-Ordering
• Section 12: Topological Spaces
• Section 13: Basis for a Topology
• Section 14: The Order Topology
• Section 15: The Product Topology on X×Y
• Section 16: The Subspace Topology
• Section 17: Closed Sets and Limit Points
• Section 18: Continuous Functions
• Section 19: The Product Topology
• Section 20: The Metric Topology
• Section 21: The Metric Topology (continued)
• Section 22*: The Quotient Topology
• Supplementary Exercises*: Topological Groups
• Section 23: Connected Spaces
• Section 24 Connected Subspaces of the Real Line
• Section 25*: Components and Local Connectedness
• Section 26: Compact Spaces
• Section 27: Compact Subspaces of the Real Line
• Section 28: Limit Point Compactness
• Section 29: Local Compactness
• Supplementary Exercises*: Nets
• Section 30: The Countability Axioms
• Section 31: The Separation Axioms
• Section 32: Normal Spaces
• Section 33: The Urysohn Lemma
• Section 34: The Urysohn Metrization Theorem
• Section 35*: The Tietze Extension Theorem
• Section 36*: Imbeddings of Manifolds
• Supplementary Exercises*: Review of the Basics
• Section 51: Homotopy of Paths
• Section 52: The Fundamental Group
• Section 53: Covering Spaces
• Section 54: The Fundamental Group of the Circle
• Section 55: Retractions and Fixed Points
• Section 56: The Fundamental Theorem of Algebra
• Section 67: Direct Sums of Abelian Groups

positron0802

Solutions to Topology, James Munkres, Chapters 2 & 3

Here you can find my written solutions to exercises of the book Topology , by James Munkres, 2nd edition. They contain all exercises from the following chapters:

• Chapter 2 – Topological Spaces and Continuous Functions,
• Chapter 3 – Connectedness and Compactness.

Unfortunately, I do not plan to write down solutions to any other chapter in the future.

This is not an official solution manual.

Chapter 2 – Topological Spaces and Continuous Functions. (Last Update: 1 January 2021.)

Chapter 3 – Connectedness and Compactness. (Last Update: 1 January 2021.)

Try to do the exercises by yourself first. Do not just copy solutions.

If you found these solutions useful and would like to support me (or want to support me even if the solutions were not useful), you can do so here:

https://www.buymeacoffee.com/positron0802

Please be aware that I wrote down these solutions while I was still an undergraduate student some time ago, so they are more likely to contain errors. Please send comments, suggestions, corrections of errors/typos, etc, by e-mail, or as a comment in this webpage.

You can mail me at [email protected]

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6 thoughts on “ solutions to topology, james munkres, chapters 2 & 3 ”.

2 unit section 20 exercise problem 9 ans pls

Thank you so much ❤

Thanks for putting the solutions out here, they are helpful in cross checking my solutions.

Thank you for your solutions. It was really helpful!

Well done！Thanks bro you’ve just save my life!

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A solutions manual for Topology by James Munkres

GitHub repository here , HTML versions here , and PDF version here .

Chapter 1. Set Theory and Logic

• Fundamental Concepts
• The Integers and the Real Numbers
• Cartesian Products
• Finite Sets
• Countable and Uncountable Sets
• The Principle of Recursive Definition
• Infinite Sets and the Axiom of Choice
• Well-Ordered Sets
• The Maximum Principle

Chapter 2. Topological Spaces and Continuous Functions

• Topological Spaces
• Basis for a Topology
• The Order Topology
• The Product Topology on X × Y
• The Subspace Topology
• Closed Sets and Limit Point
• Continuous Functions
• The Product Topology
• The Metric Topology
• The Metric Topology (continued)
• The Quotient Topology

Chapter 3. Connectedness and Compactness

• Connected Spaces
• Connected Subspaces of the Real Line
• Components and Local Connectedness
• Compact Spaces
• Compact Subspaces of the Real Line
• Limit Point Compactness
• Local Compactness

Chapter 4. Countability and Separation Axioms

• The Countability Axioms
• The Separation Axioms
• Normal Spaces
• The Urysohn Lemma
• The Urysohn Metrization Theorem
• The Tietze Extension Theorem
• Imbeddings of Manifolds

Chapter 5. The Tychonoff Theorem

• The Tychonoff Theorem
• The Stone-Čech Compactification

Chapter 6. Metrization Theorems and Paracompactness

• Local Finiteness
• The Nagata-Smirnov Metrization Theorem
• Paracompactness
• The Smirnov Metrization Theorem

Chapter 7. Complete Metric Spaces and Function Spaces

• Complete Metric Spaces
• A Space-Filling Curve
• Compactness in Metric Spaces
• Pointwise and Compact Convergence
• Ascoli’s Theorem

Chapter 8. Baire Spaces and Dimension Theory

• Baire Spaces
• A Nowhere-Differentiable Function
• Introduction to Dimension Theory

Chapter 9. The Fundamental Group

• Homotopy of Paths
• The Fundamental Group
• Covering Spaces
• The Fundamental Group of the Circle
• Retractions and Fixed Points
• The Fundamental Theorem of Algebra
• The Borsuk-Ulam Theorem
• Deformation Retracts and Homotopy Type
• The Fundamental Group of Sn
• Fundamental Groups of Some Surfaces

Chapter 10. Separation Theorems in the Plane

• The Jordan Separation Theorem
• Invariance of Domain
• The Jordan Curve Theorem
• Imbedding Graphs in the Plane
• The Winding Number of a Simple Closed Curve
• The Cauchy Integral Formula

Chapter 11. The Seifert-van Kampen Theorem

• Direct Sums of Abelian Groups
• Free Products of Groups
• Free Groups
• The Seifert-van Kampen Theorem
• The Fundamental Group of a Wedge of Circles
• Adjoining a Two-cell
• The Fundamental Groups of the Torus and the Dunce Cap

Chapter 12. Classification of Surfaces

• Fundamental Groups of Surfaces
• Homology of Surfaces
• Cutting and Pasting
• The Classification Theorem
• Constructing Compact Surfaces

Chapter 13. Classification of Covering Spaces

• Equivalence of Covering Spaces
• The Universal Covering Space
• Covering Transformations
• Existence of Covering Spaces

Table of contents

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Topology ---Homework so far

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I'm doing EVERY exercise in munkres' topology textbook

• Thread starter Tom1992
• Start date Jan 21, 2007
• Tags Exercise Munkres Textbook Topology
• Jan 21, 2007
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A PF Planet

guillemin - pollack, spivak calculus on manifolds, adams and shifrin linear algebra, my webnotes on linear algebra (14 pages of text, lots of exercises, all proofs).  

mathwonk said: guillemin - pollack, spivak calculus on manifolds, adams and shifrin linear algebra, my webnotes on linear algebra (14 pages of text, lots of exercises, all proofs).

mathwonk said: adams and shifrin linear algebra,

Why would you want to do multivariable calculus? I mean, why would you choose to use that label for what it is you want to study? It seems a bad name to use, since it implies some dull engineering nonsense. Better to stick with learning differential manifolds, if you really have to. Though I'd personally prefer to push you towards algebraic geometry rather than differential geometry. Starting with the simple book by Carter Seagal and MacDonald. It is possibly beyond where you're at now, but would be a good book to have, and is cheap. I don't understand why yo'ure complaining that the exercises have solutions on the web for spivak. Firstly, answers to almost all exercises appear somehwere on the web, and secondly, no one is making you read them. Jacobson is good (and expensive) for algebra. Again, I think you're making an error in wanting a book on linear algebra. Linear algebra is just the representation theory of a field, and that is a trivial subset of far more interesting subjects. Investing in Jacobson would set you up for a lot of pure maths. Getting anything written by Serre would be useful, too.  

ok, I've decided get my dad to order the following books for me to practice more proof exercises with linear algebra and advanced calculus: Analysis on Manifolds - Munkres Calculus on Manifolds - Spivak Advanced Linear Algebra - Roman Linear Algebra Problem Book - Halmos Linear Algebra: Challenging Problems for Students - Zhang can't wait to get them!  

here is a sample question from the munkres topology book that i find interesting (and not posted in the web). I'm going to write P for the cartesian product symbol with i taken from all of I (the index set), and U for the union symbol with i taken from all of I. let I be a non-empty index set. prove that if PAi is finite, then Ai is finite for every i in I. not difficult, but it's only interesting if you try to prove it without using i-tuples. this is more fun! here's my proof: assume that I is a non-empty index set and that PAi is finite. PAi is by definition the set of all mappings x: I -> UAi such that x(i) belongs to Ai for every i. for each x, define y as the restriction of x to i for some fixed i in I. then every y (some of which are identical to each other) is a mapping into Ai , and so Ai is the set of all the y. consequently, since all the y’s are obtained by restricting all the x’s to i, then Ai cannot have more elements than PAi and hence is finite. since i was an arbitrary element in I, then Ai is finite for every i.  

I admire you greatly. I have tried, and was never able to complete ALL the problems from any textbook. Somehow I can only work on problems I find interesting. Although, I am trying to work through every Putnam problem in history.  

A PF Asteroid

Tom, Please read https://www.physicsforums.com/showthread.php?t=8997" . LaTex is very easy to use, please learn to use it.  

A PF Organism

Yes I was going to say that too, LaTex is not as hard to use as you first think. Not to mention, it would be worthwhile because mathematicians these days are required to know some basic LaTex. Personally I've never been bothered to do all the problems from a textbook. In the trivial exercises which I believe I am strong at, I choose some sort of a pattern, depending on how many questions, and how easy they are to me. Usually its every 2nd question. For the challenging exercises I try to do them all. Several things have worried me about you Tom- First when I found out your level of Knowledge, I thought you must have been quite advanced from the start. Then I found out however, that you hadn't learned trig until you we're 11, which made me wonder how you advanced so far in the space of 3 years. Now I seem to find out that you had just been reading and not doing the exercises in what you had learned, which worries me even more. I've had similar bouts where I learned the theory, I READ the entire textbook and If someone asked me anything about the theory I could do it perfect. When it came to the questions however, It was worse than the Challenger disaster. You will find later that although you have accelerated you learning heavily, you've lost a good chunk too. Going back on subjects and re learning them from a different perspective is hard work, and would have been less if you learned it solidly from the start. I always used to think, If I know the theory good enough, then I'll be able to apply it when I need to, no sweat. That unfortunately could not be further from the truth. Basically, the gist of it is: Learn the theory the same time you do the exercises! O, and why do you seem to care so much if the solutions are posted on the web? No one is making you read them, do them yourself.  

thanks for your concerns gib. you and i both started reading calculus at an early age, me at 11 (after finishing trig in a month or so), and you at around 13 perhaps. we've both done the exercises in calculus, but when i found the exercises in calculus quite easy (it's all just calculation), i felt that i did not need to spend so much time doing exercises any more and felt I could just read through an entire textbook and learn faster. so i did this, starting with linear algebra, then over the next three years: vector calculus, differential equations (didn't like too much--too computational), groups, rings (but got bored of that), number theory, topology (which i loved, hence this thread), differential geometry/topology, and just a few days ago i decided to stop in the middle of my riemannian geometry textbook. so how much did i really learn? well, I've been diagnosed with a memory score of 150, and reading the proofs to every theorem indirectly helps one in doing proofs, and in terms of comprehension, in a few days i did all the problems in chapter 1 of munkres' topology without getting stuck, but perhaps i will get stuck later on, well see... you're right though, i should have done the problems while learning at the same time, but i just couldn't wait to learn all the topics lying ahead of me. there is just so much mathematical treasure to be had.  

What does a "memory score of 150" represents? Btw - 14 years old and all this math knowledge behind you already, I'm all but worried about you!  

so back to the discussion of this topology textbook, if anyone has any questions about an exercise in munkres' topology book, let me know and i will try to post my solution (once i get to that exercise, I'm going in order).  

Tom1992 said: ok, I've decided get my dad to order the following books for me to practice more proof exercises with linear algebra and advanced calculus: Analysis on Manifolds - Munkres Calculus on Manifolds - Spivak Advanced Linear Algebra - Roman Linear Algebra Problem Book - Halmos Linear Algebra: Challenging Problems for Students - Zhang can't wait to get them!

my dad said he'll buy me any books i want.

chris, nice to meet you. from your history of posts, i take it you are a relativity expert. the differential geometry and riemannian geometry textbooks I've read may be of some background for me to read about general relativity, if i should choose to explore there i may start doing all the exercises from a relativity book as well and post that thread in the relativity forum (but it will have to be a mathematical gr book). i'm too shy to talk to the math professors, or anyone else around me for that matter. everyone there always looks over my shoulders (literally). i only have my mathematically inclined dad to help me out. I'm turning 15 in aug 14 btw.  

as to the negative reviews of adams and shifrin by the cretins posting on amazon, one thing you need to learn is not to take the advice of students who are less intelligent than you are.  

Actually Tom, I finished Calculus when I was 11 as well, and since then I've obviously haven't learned as broad a spectrum as you. I have focused mainly on Number Theory and finding unique proofs to everything. Maybe I should have studying a broader spectrum, seeing as Number Theory require knowledge from many fields of mathematics, but I am starting that now so I am Fine. Knowing Calculus at our age is no big deal, I have numerous friends who knew it at our age, and one who knew it when she was 8. I know pretty much nothing compared to you or her, but I've enjoyed my time :)  

mathwonk said: may i suggest that if you think the reviews of adams and shifrin by the cretins posting on amazon are more reliable than advice from professors posting here, you might lose credibility as a serious student, at least with me.

• Jan 22, 2007

Tom, I have a quick question. In this learning of mathematics, is your learning strictly contained within the areas you have studied or have you worked them into a homogenous whole? Where does logic/set theory/category theory fit into your learning? I am no mathematician but think you might provide me with some insight.  

sorry for the defensive tone, please reread my edited post.  

verty said: Tom, I have a quick question. In this learning of mathematics, is your learning strictly contained within the areas you have studied or have you worked them into a homogenous whole? Where does logic/set theory/category theory fit into your learning? I am no mathematician but think you might provide me with some insight.

Tom1992 said: chris, nice to meet you. from your history of posts, i take it you are a relativity expert.

Tom1992 said: the differential geometry and riemannian geometry textbooks I've read may be of some background for me to read about general relativity,

Tom1992 said: if i should choose to explore there i may start doing all the exercises from a relativity book as well and post that thread in the relativity forum (but it will have to be a mathematical gr book).

Tom1992 said: i'm too shy to talk to the math professors, or anyone else around me for that matter. everyone there always looks over my shoulders (literally). i only have my mathematically inclined dad to help me out.

Tom1992 said: i'm turning 15 in aug 14 btw.

This kid blows my mind. Do kids of this caliber, tend to approach problems differently? I feel useless now, lol. I can't even imagine opening a topology book, let alone understanding it at my level and I am a physics major.  

Tom1992 said: gib, who is your female friend who learned calculus at age 8? if you're not already interested in her, perhaps you could introduce her to me? ;)

tehno said: You know,we all worry about you due to the fact you're already in a Riemann geometry stuff...

hrc969 said: I was sondering if anyone could explain why this would cause worry about him. ... Can someone please explain?

tehno said: A nut house before he turns 18?

Well I would rather see that someone who finds enjoyment in intellectual pursuits continue doing that than try to do what everyone else does. If Tom has a gift then why should he not pursue it? Moreover, I'm pretty sure it's up to him to decide whether chasing girls is a worthy life purpose.  

A PF Molecule

I, and I assume some of the people here that have read this thread, just want to say congratulations , and that I'm jealous :) Not of your intelligence; I'm fairly intelligent myself as are most people in my field. I'm really just jealous of your motivation/work ethic and your introduction to these topics at such an early age. I would love to sit down and just start studying new areas of mathematics or physics that I haven't yet learned. Unfortunately I have so much going on in my life (Preparation for grad school, looking for work, girlfriend, etc) that I find it hard to sit down and study as a means of relaxation. I'd rather read a novel. I'm happy you've been able to sate this interest and hope that while doing so you haven't hindered any of your creative abilities. Make sure you read imaginative works as well and help develop your creative side. Just knowing everything about a field of mathematics will not enable you to develop new ideas and theories. And just on a side note, you said Differential Equations had too much calculation. Were you speaking of algebraic/calculatory solution, or actual numeric calculations. I don't recall DiffEq having a single numeric solution when I was taught, and I think that both it and Boundary Value Problems are immeasurable tools for solving both real world and theoretical problems. It provides you with a very straight-forward analytical viewpoint on virtually any dynamic process. (Circuits, Heat, Solids, Thermodynamics/Stat Mech, Dynamics, Quantum, etc). And nearly anything that has vibrational motion uses DiffEq/BVP methods for solution. But perhaps you're past all of that already. I do not know. But I wish you well, and good luck! -KJ Healey  

please understand that i am only the youngest student in my first year classes, but probably not the smartest.  

please understand that i am only the youngest student in my classes, not necessarily the smartest, otherwise i might actually speak out in class.

here is another interesting munkres problem from chapter 1. i will repeat once more that all problems i post in this thread do not already have a solution posted in the web. prove that the set A = {a,b}x{a,b}x{a,b}x... is uncountable. the solution is very short, but requires deep thinking. proof: suppose that A is countable. then there is a surjective map f:N -> A, where N is the set of positive integers. define x as the unique element in A whose nth coordinate is different from f(n) for all n in N (this construction is possible since each component has two possible values). since f is surjective, then there exists m in N such that f(m)=x. but this is impossible since by construction x differs from f(m) in the mth component. this contradiction means that A cannot be countable. this same trick can be used to prove that the real numbers is not a countable set, using the infinite decimal repesentation of a real number. try it!  

If P is a countable subset of A with cardinality omega (equal to the cardinality of omega), I'm wondering what the cardinality is of the set of surjective maps from N onto P...  

my reading has now actually slowed down. whenever i read a theorem i glance at how long the proof is. if it is short, i try to come up with the proof myself. usually such short proofs to theorems serve as valid a problem as the textbook problems. so i guess now it will take me longer to finish this book than i expected, but time is on my side. however, i don't always manage to come up with the proof: some short proofs are sooooo hard! for example: prove that there is no surjective map f: A-> P(A), where P(A) is the set of all subsets of A. proof: define B={a in A: a belongs to A-f(a)}. suppose B=f(c). then [c belongs to B] iff [c belongs to A-f(c)] iff [c belongs to A-B], a self-contradiction. wow!  

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History of Stavropol Krai

The most ancient archaeological finds date back to the 4th millennium BC. The territory of the present Stavropol region was successively part of the state of the Scythians (the 7th - 5th centuries BC), Sarmatians (the 3rd century BC - the 3rd century AD), Huns (the 4th - 5th centuries AD).

Later, from 620 to 969, this territory was part of the ancient state called the Khazar Khaganate. Approximately in the 8th century, with the weakening of the Khazar Kaganate, the medieval state of the Alans appeared here. In 1238-1239, a significant part of the plain Alania was captured by the Mongols, and this state as a political entity ceased to exist.

In 1556, the Russian troops took Astrakhan and opened the way to the North Caucasus and the Caspian Sea. In Ciscaucasia, the interests of Russia, the Ottoman Empire, the Crimean Khanate, and Iran collided.

In 1777, according to the decree of Catherine II, the Azov-Mozdok defensive line was founded, which gave rise to colonization of the Ciscaucasia and the North Caucasus. The territory of the Stavropol region became part of Astrakhan oblast. In November 1777, the fortress called Stavropolskaya was founded. In 1782, about 500 retired soldiers lived there.

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In 1785, in connection with the development of Ciscaucasia, the Caucasian guberniya (province) was created that included the Caucasian and Astrakhan regions. Since that time, Stavropol officially became one of the six county-level towns of the Caucasus region.

With the development of the Ciscaucasia, Stavropol was gaining an increasing importance as an important trade and transit center. It became a kind of the main gate of the Caucasus. In 1822, the Caucasian province was transformed into an oblast and Stavropol became its center. After the defeat of the Decembrist uprising, a lot of its participants were sent here. In 1837 - 1841, Mikhail Lermontov, exiled to the Caucasus, visited Stavropol several times.

In 1847, the Caucasian oblast was reformed into Stavropol gubernia. With the formation of the Kuban and Terek Cossack regions and the end of the Caucasian War, the military-political and economic importance of Stavropol significantly reduced.

In 1919, the Stavropol province was occupied by the Bolsheviks and included in the territory of the North Caucasian Soviet Republic. As a result of the Second Kuban campaign the region went under the control of the Volunteer Army.

In October 1924, the North Caucasian region was formed and Stavropol gubernia was reformed into a district within the region. On January 10, 1934, the North Caucasian Krai was divided into the Azovo-Chernomorsky and North Caucasian. The town of Pyatigorsk became the center of North Caucasian Krai. In March 1936, North Caucasian Krai was reformed and, on its territory, Ordzhonikidze Krai with the center in Ordzhonikidze (Stavropol) was formed.

During the Second World War, from August 1942 to January 1943, the region was occupied by the German troops. In 1943, Ordzhonikidze Krai was renamed Stavropol Krai. In December 1956, the first part of the Stavropol-Moscow gas pipeline with a length of 1,300 km was commissioned (at that time, it was the longest gas pipeline in Europe).

During the 1970s-1980s, 56 new enterprises were opened in the region, among them the Prikumsky Plastics Plant - the largest chemical plant in the region, four power units at the Stavropol power station, and new capacities at the Nevinnomyssk enterprise “Azot”.

On July 3, 1991, Karachay-Cherkess Autonomous Region withdrew from Stavropol Krai and became the Karachay-Cherkess Soviet Socialist Republic. On April 21, 1992, it became the Republic of Karachay-Cherkessia of the Russian Federation.

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Stavropol Krai stretches for 285 km from north to south and 370 km from west to east. The climate is temperate continental. The average temperature in January is minus 5 degrees Celsius (in mountains - down to -10), in July - plus 22-25 degrees Celsius (in mountains - +14).

The main natural resources are natural gas, oil, polymetals containing uranium, building materials. Mineral medicinal waters are a special riches of the region.

The Caucasian Mineral Waters is Russia’s largest resort region, which has no analogues in the whole of Eurasia for the richness and diversity of mineral waters and therapeutic mud. The healing properties of “narzan”, one of the popular local mineral waters, are known throughout Russia. The name can be translated into Russian as “Hercules’ beverage”, “Water of Hercules”.

The largest cities and towns are Stavropol (458,200), Pyatigorsk (145,500), Kislovodsk (127,300), Nevinnomyssk (114,400), Yessentuki (117,200), Mikhailovsk (94,500), Mineralnye Vody (72,400), Georgievsk (64,400), Budennovsk (59,600).

Stavropol Krai - Economy

The main industries of Stavropol Krai are engineering, production and processing of oil and natural gas, electric power industry, food (winemaking, butter, sugar), chemical (mineral fertilizers in Nevinnomyssk), building materials (glass in Mineralnye Vody), light (wool in Nevinnomyssk, leather in Budennovsk).

Agriculture specializes in growing grain and sunflower, the leading role in livestock breeding belongs to cattle breeding, fine-wool sheep breeding. Horticulture, viticulture, poultry farming, pig breeding, beekeeping are widespread. Agriculture is one of the most important sectors of the local economy, which employs more than 156 thousand people.

The main highway M29 “Caucasus” passes through Nevinnomyssk, Mineralnye Vody and Pyatigorsk. There are international airports in Stavropol (Shpakovskoye) and Mineralnye Vody. This region has a very dense and extensive network of pipelines.

Attractions of Stavropol Krai

A large number of various interesting places are concentrated on the territory of the Stavropol region. Here are just a few of the most famous sights:

• Proval - a lake and a natural cave on the southern slope of Mount Mashuk in Pyatigorsk. The cave is a cone-shaped funnel with a height of 41 m, at the bottom of which there is a karst lake of mineral water of pure blue color;
• Monument to Lermontov in Pyatigorsk at the place where the poet was fatally wounded during the duel;
• Lake Tambukan (Black Lake), located near Pyatigorsk, is known for its unique healing mud;
• Therapeutic park, mineral springs, Balneary mud baths named after Semashko in the resort city of Yessentuki;
• Resort park in Kislovodsk is very popular with tourists. The territory of the park is huge. Here you can find a drinking gallery, ponds, grottoes, and the famous valley of roses. Plants growing in the park make the air unusually clean and healthy;
• Koltso (Ring) Mount near Kislovodsk. Under the influence of natural factors, a ring with a diameter of 8 meters was formed in the center of the rock;
• Pushkin Gallery (1901), the Emir of Bukhara Palace, the Cave of Permafrost, Zheleznaya Mount in the resort town of Zheleznovodsk.

Stavropol krai of Russia photos

Stavropol Krai scenery

Paved road in Stavropol Krai

Author: A.Kostin

Winter in Stavropol Krai

Author: Kabatov V.

Small river in the Stavropol region

Author: Alex Stanin

Pictures of Stavropol Krai

Beautiful nature of Stavropol Krai

Author: Sergey Shevchenko

Author: V.Buturlia

Cathedral in Stavropol Krai

Author: Bulgakov Pyotr

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